We're talking about different things. I'm talking about formal theories and interpretations of their languages as discussed in mathematical logic, and such that theories are not interpretations. — TonesInDeepFreeze

But, just to be clear, we still need to prove that there exists a set such that every natural number is a member of that set, since that set is the domain of the aforementioned sequence. — TonesInDeepFreeze

You don't know enough — TonesInDeepFreeze

The crank — TonesInDeepFreeze

Making clear corrections, giving generous explanations, and posting ideas in general is not ranting. — TonesInDeepFreeze

And by starting with Enderton's logic book, which does present the axioms for '=', you would see how they work in set theory even if not explicitly stated in his set theory book. — TonesInDeepFreeze

But when you complained that it does not mention identity theory, I said that I would have been mistaken if I offered it for reference on that matter. And, now that I see the context, I grant that, since the context was general, it would not be entirely unreasonable for you to take it that at least part of the reason for my recommending the book is that it mentions identity theory, so, in that respect, my recommendation my be faulted. — TonesInDeepFreeze

But then I followed up by pointing to Enderton specifying the equality axioms in his logic book (though he doesn't mention in that book the fact that set theory is based on first order logic with equality). And that was pertinent to your complaint that you couldn't find anything on that topic. — TonesInDeepFreeze

And I cited Hinman's book that both gives the axioms for equality as part of first order logic, equivalent to the axioms I posted, and he says that set theory is based on first order logic. — TonesInDeepFreeze

And I referred you to Shoenfield's book that specifies the axioms for '=', equivalent to the axioms I posted. — TonesInDeepFreeze

And, you yourself agree that set theory is based in first order logic. — TonesInDeepFreeze

So, all that is needed is to show citations that first order logic ordinarily includes identity theory (i.e. first order logic with equality) and that was accomplished by citing Enderton's logic book, Hinman, and Shoenfield. — TonesInDeepFreeze

But I guess that, despite my sin of overlooking that a certain book doesn't supply reference to a particular point (though it still is an excellent reference for the context of this subject and on other points) it seems I am finally past needing to explain over and over and over that the identity axioms are in first order logic and set theory is based in first order logic, as you post: — TonesInDeepFreeze

Giving pinpoint corrections, copious explanations, and sharing ideas in general is not ranting — TonesInDeepFreeze

In that instance, yes, and made clear by what I wrote. — TonesInDeepFreeze

Of course, that's hardly even a foible. But it's at least odd that someone who knows nothing about the matter would categorically say that it false that the indiscernibility of identicals is not included in first order logic with '=' as primitive. — TonesInDeepFreeze

We were talking about how '=' is interpreted. — TonesInDeepFreeze

Now why I'm ranting so much about negative self-reference or diagonalization, which I acknowledge I haven't accurately defined, is that it crops so easily in many important findings. Yet what is lacking is a general definition. — ssu

What I meant that it itself is an indirect proof: first is assumed that all reals, lets say on the range, (0 to 1) can be listed and from this list through diagonalization is a made a real that is cannot be on the list. Hence not all the reals can be listed and hence no 1-to-1 correspondence with natural numbers. Reductio ad absurdum. — ssu

The reason why physical collections are different from sets, in this way, is that physical objects are different from intelligible (including mathematical) objects. What I am concerned about is that the law of identity, as formulated from Aristotle, is specifically designed from a recognition of this difference, and intentionally designed to protect, and maintain the understanding and acceptance of that difference. To put it simply, an abstraction, intelligible object, is a universal, and a physical object is a particular. The law of identity refers to the identity of a particular. And, because intelligible objects are different from physical objects, as you recognize and acknowledge, they cannot be held to this law. So mathematical ideas, if they are called "objects", are objects which naturally violate the law of identity. In short, that's how we distinguish physical objects from ideas, with the law of identity. — Metaphysician Undercover

In classical sophistry physical objects are confused, mixed up, and conflated with intelligible objects. The difference between the particular and the universal, as "objects" is ignored. This allows sophists to logically prove things which are absurd. The law of identity is intended to enforce that difference, and expose the faults of the sophist. The head sophist at TPF, TIDF, continues to defend sophistry by arguing that intelligible objects are consistent with the law of identity. — Metaphysician Undercover

I'd have to say, no, not really. Internal/external properties is a distinction we make concerning the properties of particular physical objects, the object's internal relations, and the object's external relations. Intensional/extensional meaning is a distinction concerning the meaning of a word, how the word relates to ideas, and possibly physical objects. This is a matter of semiotics, and Charles Peirce provides some very good insight into the use of symbols. But that is a completely different matter from what I was discussing, as the internal/external properties of a physical object. — Metaphysician Undercover

The problem, is that you continually cross the boundary of separation between physical objects, and intelligible objects, in your manner of speaking, in the sophistic way, without even noticing it. — Metaphysician Undercover

That's what happened with your example of schoolkids. In order for the example to work, "schoolkids" must refer to a multitude of particular physical objects. Yet "set" must refer to an intelligible object. So in speaking the example you cross the category separation, back and forth in the way of sophistry, without even realizing it. — Metaphysician Undercover

Imagine if we were to maintain the boundary. Instead of having schoolkids in a playground, we would be talking about the idea of "schoolkid", or an imaginary schoolkid. This appears to deny the possibility of any extensional meaning. Further, if we want a number of schoolkids, then we need a principle of separation to distinguish one from the other. But that principle of separation would either create an order amongst the imaginary schoolkids, or else produce a complete separation of type, making distinct types of schoolkids. — Metaphysician Undercover

OK, you have no interest in the difference between a subject to be studied and an object to be studied. — Metaphysician Undercover

That's fine by me, but until you learn this difference you are likely to continue to speak in a way which mixes these two up, and makes your examples and arguments appear like nothing more than sophistry, and arguing by equivocation, just like Tones. — Metaphysician Undercover

This is what happens when a subject is called an object (mathematical) and the difference between the physical object and the mathematical object, (as defended by the law of identity) is ignored. — Metaphysician Undercover

That's right, you are not my philosophy professor, that would reverse credentials. I am your philosophy professor, and your lack of interest deserves a failing grade. — Metaphysician Undercover

Right, this is why a set is not an object, objects have internal properties and external properties, sets have meaning. — Metaphysician Undercover

There is no "instance" of any set. — Metaphysician Undercover

You recognize that there is a difference between physical objects an sets, why do you not see that there is no such thing as an instance of a set? — Metaphysician Undercover

Sets are not the type of thing which have an instantiation. "Instance" refers to a particular, a set is a universal. That sort of misleading statement is where the sophistry kicks in, even though I know you are not intending to be misleading.. — Metaphysician Undercover

That's a simple question with a simple answer. When a rule in a game contradicts another rule in a game, this is cause for disbelief in the whole game. That was the point of the example I gave you of waves in physics. — Metaphysician Undercover

That has become obvious to me. But in a philosophy forum, things ought to be the other way around. We ought to be discussing the ontology of sets and working through the problems which arise. — Metaphysician Undercover

There's too many concerns to summarize. But let's look at a most fundamental problem of set theory as an example. You recognize the difference between physical objects, and sets, so let's start there. — Metaphysician Undercover

Now, consider the elements of a set, these might be sets as well. — Metaphysician Undercover

The elements of a set are not physical objects, just like sets are not physical objects. — Metaphysician Undercover

The elements are ideas, universals, they are not particulars or individuals. — Metaphysician Undercover

Since they are not particulars the set cannot be measured as particulars. A set cannot have a cardinality. That's a basic problem. — Metaphysician Undercover

Please allow me to respond in the context of the SB-tree. Fractions correspond to nodes. Reals correspond to arbitrarily long paths (well, almost but providing clarifying details would bloat this post). There's no point to introduce natural numbers, integers, or rational numbers as disagreement would ensue. I would say they are all fractions but you would likely say they are all reals. — keystone

The cut at fraction 1/1 is fully captured at row 1 of the tree. — keystone

The algorithm corresponding to the cut at real 1.0 generalizes how the cut would be captured at any arbitrary row beyond row 1 (well, to be precise I should really use ε_left and ε_right instead of just ε). Finally, the execution of the cut at 1.0 happens on a particular row once the computer chooses values for the ε's. What should be clear is that none of this happens at the bottom of the tree. This is an entirely top-down approach. — keystone

I think it's just that Latex does not get used properly in quotes. — keystone

I'm rewriting my last post in plain text and using the notation I recently proposed.
---------------------------------------------------------------------------
I'd like to distinguish between a fraction and a real. The fraction description is finite (e.g. 1/1), whereas the real description is infinite (e.g. 1.0, which can be represented as an algorithm that generates the Cauchy sequence of fractional k-intervals: <9/10, 11/10>, <99/100, 101/100>, <999/1000, 1001/1000>, <9999/10000, 10001/10000>, ...

Because fraction descriptions are finite, a cut at a fraction can be planned and executed all in one go. A cut of <-∞, +∞> at the fraction 1 results in: <-∞, 1> ∪ 1 ∪ <1, +∞>.

Because real descriptions are infinite, a cut at a real must be planned and executed separately.

The algorithm to cut <-∞, +∞> at the real 1.0 is generalized as: <-∞, 1-ε> ∪ 1-ε ∪ <1-ε, 1+ε> ∪ 1+ε ∪ <1+ε, +∞> where ε can be an arbitrarily small positive number.

In the spirit of Turing, the execution of the cut of <-∞, +∞> at the real 1.0 could have us replace ε with 1/10 as follows: <-∞, 9/10> ∪ 9/10 ∪ <9/10, 11/10> ∪ 11/10 ∪ <11/10, +∞>
---------------------------------------------------------------------------
One thing that I've failed to get across is that I'm not outlining a procedure which will be used to construct infinite numbers. These systems I'm outlining, such as <-∞, 1> ∪ 1 ∪ <1, +∞>, are valid systems in and of themselves. Finite systems such as these are all we can ever construct in the top-down view. — keystone

eiπ+1=0 — fishfry

I think the problem is precisely that there is nothing to constrain the lamp and we want to find something. In theory, we could stipulate either - or Cinderella's coach. But we mostly think in the context of "If it were real, then..." Fiction doesn't work unless you are willing to do that. It's about whether you choose to play the game and how to apply the rules of the game. — Ludwig V

This seems to be more in tune with common sense, for what it's worth. The question is, why? I think it is because of the dressing up of the abstract structure. We assume the lamp has existed before the sequence and will continue to exist after it. — Ludwig V

So the fact that the sequence does not define it does not close the question and we want to move from the possible to the actual. But it is not clear how to do that - and we don't want to simply stipulate it. Perhaps that's because defining the limit of the convergent sequence as 1 - or 0, which have a role in defining the sequence in the first place, — Ludwig V

invites us to think in the context of the natural numbers (or actual lamps), whereas defining ω as the limit of the natural numbers does not. — Ludwig V

This doesn't make sense. Each flip of the coin is an individual act, and it has a single outcome. Once the outcome is achieved, that outcome stands until there is another flip. The outcome "can't be both at different times", because a different outcome requires a different flip. However, there can be different outcomes from different flips. — Metaphysician Undercover

That was a complete description. — Michael

There are no hidden assumptions. — Michael

P1 is implicit in Thomson's argument. Using the principle of charity you should infer it. As neither you nor Benacerraf have done so I have had to make it explicit. — Michael

The lamp is off at 10:00. The button is pushed 10100100 times between 10:00 and 10:01. Is the lamp on or off at 10:02? — Michael

Any reasonable person should infer that nothing else happens between 10:01 and 10:02. — Michael

Even though this is a physically impossible imaginary lamp, and even though I haven't told you what happens at 10:02, it is poor reasoning to respond to the question by claiming that the lamp can turn into a plate of spaghetti. The correct answer is that because 10100100 is an even number, the lamp will be off at 10:02. — Michael

There is no Supreme Button Pusher arbitrarily willing the lamp to be on or turning it into a pumpkin. There is only us pushing the button once, twice, or an infinite number of times, where pushing it when the lamp is off turns the lamp on and pushing it when the lamp is on turns the lamp off. — Michael

I think you should distinguish the Democrat voters with whoever's running the DNC. The Democrats by and large didn't want Biden to run in 2024 and the DNC as usual didn't listen. — Mr Bee

I post for at least as an end in and of itself, and also meaningful record for whomever may read it, no matter how few people or even accepting that it might be none at all. It would be good if my best efforts in explanation were understood, but I cannot ensure that they are, especially given that they are ad hoc and out of context of the required material they depend on. — TonesInDeepFreeze

We're going around full circle. — TonesInDeepFreeze

(1) I said it may be more commonly called 'first order logic with equality'.

(2) For about the fourth time, a only a few posts ago I gave the axioms. And you responded by asking why I posted it! — TonesInDeepFreeze

(3) And I gave you a reference to Enderton where he stated an axiomatization equivalent with the one I gave. And Hinman also, and moreover as he states set theory as based on first order logic (which is to say, first order logic with equality). — TonesInDeepFreeze

(4) You said yourself that you recognize that set theory is based on first order logic. Set theory is based on first order logic with equality. That is what identity theory is, as I've said before. Whether called 'identity theory' or 'first order logic with equlality', it's the same set of axioms. — TonesInDeepFreeze

Yes, because the reasons I mentioned go the heart of the motivation for the axioms. — TonesInDeepFreeze

That's up to you. But I am not errant for correcting things that are wrong. — TonesInDeepFreeze

And you studied with Shoenfield. On page 21 lines 13 and 15 of his book you will see the equality axioms that are the indiscernibility of identicals, similar to the way I formalized and that you asked why I posted it. — TonesInDeepFreeze

So what? In logic it is ordinarily stipulated. — TonesInDeepFreeze

I don't see it as a confusion of Michael. He is only rendering Thomson's setup. And I don't see Michael getting tripped up by the metaphorical use of a lamp and button. And I don't see Thomson as getting tripped up either. — TonesInDeepFreeze

Moreover, they seem to interfere sometimes when people get hung up on how to relate such a hypothetical lamp and button with actual lamps and buttons. — TonesInDeepFreeze

I don't recall the context in which I recommended Enerton's set theory book, but if it was about first order logic with identity for set theory, then I mis-recommended. — TonesInDeepFreeze

Who was the famous logician? — TonesInDeepFreeze

Shoenfield's logic textbook is rich and has lots of stuff not ordinarily in such a book. But it is difficult, and he uses some terminology inconsistent with ordinary use in the field. — TonesInDeepFreeze

As I recall, many posts ago, my initial point was that, contrary to your assertion, the axiom of extensionality, as ordinarily given, is not a definition. — TonesInDeepFreeze

Then I went on to explain how there are other ways to set up the logic and the set theory axioms so that a different version of extensionality would be a definition. — TonesInDeepFreeze

An ordinary presentation of set theory either explicitly or implicitly has set theory based upon first order logic with identity theory. — TonesInDeepFreeze

Yes, of course, set theory has non-logical axioms, so set theory is not just first order logic with identity. — TonesInDeepFreeze

if A and B are both sets
use extensionality from set theory
else
use identity from logic
— fishfry

That's not right. — TonesInDeepFreeze

In set theory, we use both the logic axioms (which include the identity axioms) and the set theory axioms (which include the axiom of extensionality). lf our focus now speaking is identity theory and the axiom of extensionality, then it suffices to say that we use both. — TonesInDeepFreeze

I don't know how I could be more clear about that. — TonesInDeepFreeze

I was trying to solve the problem that you had not been understanding me as you characterized my point again incorrectly, so I tried to state it in as simple terms as I could. — TonesInDeepFreeze

I didn't say that you did. Rather you gave your reason that we need identity. And I take it that 'identity' in that context is short for 'the axioms and semantics regarding identity', and I gave better reasons that we need them. — TonesInDeepFreeze

However, several posts ago you did indicate (as best I could tell) that you think the axiom of extensionality is all we need for proving things about identity in set theory, — TonesInDeepFreeze

which would comport with your view that the axiom of extensionality is a definition. — TonesInDeepFreeze

So, I mentioned that, for example, from the axiom of extensionality alone we cannot prove:

(x = y & y = z) -> x = z — TonesInDeepFreeze

We need the law of identity, but we also need the indiscernibility of identicals. — TonesInDeepFreeze

(But Wang has an axiomatization in a single scheme.)

Yet, interestingly, from the axiom of extensionality we can derive the law of identity:

(1) Az(z e x <-> z e x) logic

(2) x = x from (1) and the axiom of extensionality

But the law of identity does not ensure that '=' stands for an equivalence class. It only provides

x = x

It does not entail

x = y -> y = x

nor

(x =y & y = z) -> x = z

To get all the needed identity theorems, we need both the law of identity and the indiscernibility of identicals. — TonesInDeepFreeze

Moreover, we want to ensure that '=' stands not just for an equivalence relation but for, indeed, the identity relation. — TonesInDeepFreeze

And to do that we have to make the semantic stipulation that '=' is interpreted as standing for the identity relation on the universe. — TonesInDeepFreeze

Me too. There is so much I didn't learn a long time ago but should have learned. I never got past a pretty basic level. And now I am very rusty in what I did learn, and don't have very much time to re-learn, let alone go beyond where I was a long time ago. — TonesInDeepFreeze

You said that sets have sets as members and that there is a pickle about that viv-a-vis identity. — TonesInDeepFreeze

Of course it is not. — TonesInDeepFreeze

I'd like to distinguish between a fraction and a real. The fraction description is finite (e.g. 11 — keystone

which can be represented as an algorithm that generates the Cauchy sequence of fractional intervals: (910,1110),(99100,101100),(9991000,10011000),(999910000,1000110000),… — keystone

(
1
,
+
∞
)
. — keystone

eyes glazed? — keystone

I haven't made a point yet. I just wanted to clarify this as I previously found it a stumbling block in our conversation. — keystone

I'm a computational fluid dynamics analyst, so I naturally approach things from a simulation perspective. — keystone

In the context of a potentially infinite complete tree, to me it makes sense to talk about potentially (countably) infinite nodes and potentially (uncountably) infinite paths. In this sense, the paths have more potential than the nodes. I don't think any beauty is lost in reducing infinity to a potential. — keystone

As QM suggests, something funny happens when we're not observing the world. I consider this to be the magic of potential. Anyway, this is fluffy talk about potential...let me get to the beef. — keystone

The claim was directed at your example of choosing a direction at a fork in the road. The only way that you could have multiple possible outcomes is by assuming a principle that overrules the rules, i.e. transcends the rules. Freedom of choice, allows you to choose rather than follow a rule. If your example is analogous, then multiple possible outcomes being consistent with the rules, implies that choice is allowed, i.e. the rules allow one to transcend the rules. Strictly speaking the actions taken when the rules are transcended are not consistent with the rules, because these actions transcend the rules. The rules may allow for such acts, acts outside the system of rules, but the particular acts taken cannot be said to be consistent with the rules because they are outside the system. — Metaphysician Undercover

That is a fair understanding of my point but I do want to highlight one thing: it's not always about the computation. If I want to focus on algorithm design (and not execution), I can keep ε's floating around. The ε's only need to be replaced when I execute the algorithm and perform the computation. Fair? — keystone

Yes, actual infinities are beyond computation. — keystone

It does seem to be a bit magical. I'd like to avoid magical thinking if at all possible. — keystone

Enderton's set theory text is a great book. But, as with many excellent set theory books, it doesn't mention all the technical details. — TonesInDeepFreeze

I didn't say that identity is implicitly in extensionality, whatever that might mean. — TonesInDeepFreeze

I've said that usually set theory is based on first order logic with identity. That includes the identity axioms (such as found in Enderton's logic book). Then set theory adds the axiom of extensionality that provides a sufficient condition for identity that is not in identity theory. — TonesInDeepFreeze

if A and B are both sets
    use extensionality from set theory
else 
    use identity from logic

I don't know how I could be more clear about that. Explicity:

Start with these identity axioms:

Ax x=x (a thing is identical with itself)

and (roughly stated) for all formulas P(x): — TonesInDeepFreeze

We need identity axioms to prove things we want to prove about identity, including such things as: — TonesInDeepFreeze

Suggestion: Learn the details of the axioms and rules of inference of first order logic with identity. Then start with the very first semi-formal proofs in set theory (such as a set theory textbook usually gives semi-formal proofs), and confirm how those proofs would be if actually formalized in first order logic with identity. Then you would see how the axioms and rules of inference of first order logic with identity play a crucial role in set theory. — TonesInDeepFreeze

I have no idea what pickle you see. — TonesInDeepFreeze

Perhaps this is what you're trying to explain to me.

Is it?
— fishfry

No. — TonesInDeepFreeze

If you read again the first post in this thread on this particular subject, with regard to exactly what I've said, step by step, then it may become clearer for you. But also, as mentioned, learning the axioms and inference rules of first order logic with identity would be of great benefit. My suggestion would be to start with:

Logic: Techniques Of Formal Reasoning - Kalish, Montague and Mar
(but you could skip it if you feel strong enough already in doing formal proofs in symbolic logic and making simple models for proofs of consistency and of proofs invalidity by counterexample)

Then:

A Mathematical Introduction To Logic - Enderton — TonesInDeepFreeze

I found both of those books to be a special pleasure and profoundly enlightening. The Enderton book especially blew my mind, as I saw in it how mathematical logic so ingeniously, rigorously and elegantly gets to the heart of the fundamental considerations of logic while making sure that no technical loose ends are left dangling. — TonesInDeepFreeze

I'll try to do a better job this time. But first, one other area of confusion has been the distinction between infinite and arbitrary as it relates to an algorithm's design vs. it's execution. A Turing algorithm for constructing N, is designed to output a set of m elements, where m can be arbitrarily large. By this I mean that the algorithm itself sets no limit on the size of its output; rather, the size of the output depends on the execution (i.e., the chosen 'precision' based on available resources). Please note that I'm not saying that m is a particular number, nor is it infinity. Instead, when talking about the algorithm itself, m serves as a placeholder for a value that is determined only upon execution of the algorithm. Upon execution, m is replaced with a natural number and the output is a finite set. In a similar vein, when I speak of ε being arbitrarily small, I am using it as a placeholder to describe an algorithm. Upon executing that algorithm, ε is replaced with a positive rational number that is small, but by no means the smallest. Is this clear? — keystone

I think we have to look at context here. What is our subject of discussion, what are we talking about here? — Metaphysician Undercover

Are we talking about things (individuals), of which there is a multitude, or are we talking about a group (set) of individuals, of which there is one? Your description above, seems to imply the former. You are talking about separate things, many schoolkids, and there is many possibilities as to the order they could have. On the other hand, if you were talking about the group as a whole, as your subject, then the parts of that group, the individuals, must have the order that they have at that time, even though it could be different in past or future times. If you were talking about the same individuals in a different order, this would require a change to that specific group, so you would be talking about that group, at a different time, because you'd be talking about the individuals, changing places. — Metaphysician Undercover

You might understand this better through what is known as internal and external properties. To each individual, as a subject, its relations to other individuals are external properties. To the group, as a unit, and the subject, the relations between the individuals is an internal property. — Metaphysician Undercover

You talk about the schoolkids as distinct individuals, where the various relations between them are the external properties of each and everyone of them. There are no internal relations here. Each schoolkid is a subject to predication, age, height, etc.. and you might produce an order according to those predications. The order is external to each schoolkid, people say it transcends, and changing the transcendent order does not change any of the schoolkids in anyway. — Metaphysician Undercover

Now, let's take the group as a whole, as an object, and produce a corresponding subject, the set, and make that our subject. Since the whole group is our object of study, any change to the order of the individuals is an internal change to that object, therefore a change to that object itself. The order of the individuals (as the parts of the whole) is an internal property of that object, and a change to that order constitutes a change to the object, which we must respect as predicable to the corresponding subject. Therefore we can say that the order of the individuals, as the parts of the whole, is an intrinsic property of the whole, which is represented as the set. — Metaphysician Undercover

Notice however, the switch from "subject" to "object", and this I believe is the key to understanding these principles. There is an implicit gap, a separation, between the meaning of "logical subject" and "physical object". When we make a predication, "the sky is blue" for example, "the sky" is the subject, and if there is an object which corresponds with that subject, the predication may be judged for truth. However, we can manufacture subjects and predications with complete disregard for any physical objects, and so long as we have consistency, we have a valid "subject", with no corresponding object. — Metaphysician Undercover

Consider the following proposition, "There is a group of schoolkids". We have a propositional subject, without a corresponding object, what some people would call "a possible world". Since there is no assumed corresponding object which would cause a need for conformity, we can predicate any possible order we want, so long as it is not contradictory. The hidden problem of formalism which I referred to lies in the naming of the group, "schoolkids". That name needs to be clearly defined and the definition will place restrictions on what can be predicated without contradiction. — Metaphysician Undercover

Perhaps, we can remove these restrictions, by making the things within the group, the elements of the set that is, completely nondescript. "There is a group of nondescript things". We still have the name "things", with implied meaning, so this name has to be defined, and this would put restrictions on what we can predicate. So we go to a simple symbol, "x" for example, and assume that the symbol on its own, has absolutely no meaning, and this would allow any individual predication whatsoever without any risk of self-contradiction. X is a subject which has absolutely no inherent properties. — Metaphysician Undercover

It might appear like we have resolved the problem in this way, we have a subject "x" which can hold absolutely any predication, so long as the predications don't contradict.. However, when we assume that the subject has no inherent properties, we disallow any predication because the predication would be a property and this would contradict the initial assumption. So this starting point allows no procedure without contradiction. — Metaphysician Undercover

Now look what happens when we say "there is a group of x's". There is actually something implied about x, which is implied simply by saying that there is a group of them. It is implied that x has a boundary, separation, etc.. We may start with the assumption that there is no intrinsic properties of X, but as soon as we start to predicate, we negate that assumption. And the symbol, x, without any predications is absolutely useless. — Metaphysician Undercover

I agree, what I meant is that this appears to be the inherent order, but it's not necessarily, that's why I went on to say that we can deny that order. — Metaphysician Undercover

I think so, but I also think, that sort of inherent order has minimal effect, and the real issue comes up with the restrictions, or limitations to order which are constructed. What I am arguing is that how the inherent order manifests, is as a limitation to the order which one can select. If there is absolutely no inherent order, then we can select any order, but if there is limitations to what can be selected, we cannot choose any order. — Metaphysician Undercover

The examples you give are, I believe, selected, therefore they're probably no true inherent order. The example I gave, is that we cannot give 2 and 3 the same place in the order, they cannot be equal, so we need to proceed toward understanding how this limitation exists. — Metaphysician Undercover

So this is where the real problem lies, in defining a symbol, such as 2 or 3, as a set. — Metaphysician Undercover

Check back to what I said about the difference between internal and external properties. The subject now is a set, say 2, and a set necessarily has internal properties. We have the elements which compose the set, 0,1, which are also sets. As the set is also related to other sets, it has external properties, represented by the ∈
operator. The external properties are not necessary, and are a matter of choice, but whatever choice is made, that choice dictates the nature of the internal properties. — Metaphysician Undercover

Now here's where I think the illusion lies. A set necessarily has internal properties, even though there may be infinite possibility as to the nature of the internal properties, making the specific nature of the internal properties dependent on choice, in this case von Neumann's choice. The illusion is that since the specific nature of the internal properties is dependent on a choice from infinite possibilities, it would therefore be possible to have a set with no internal properties. Clarification of the illusion implies that a set cannot exist prior to the choice of external properties, which dictate the internal properties. Internal properties are essential to "a set", and so a set has no existence prior to the choice of external properties, which determine the internal properties. This makes the empty set, as a set with no internal properties, impossible. The problem now, is what is zero? It can't be a number, because numbers are sets, and an empty set is impossible. — Metaphysician Undercover

I think you misunderstand. As I explain above, you refer exactly to the internal (intrinsic) properties of 2 and 3, as sets, — Metaphysician Undercover

to show that they are different numbers. What the set theory has done is denied order as an external property of those things, 2 and 3, as numbers with order relative to other numbers, and made it into an internal property of those things, as sets. An internal property is an intrinsic order. The fact that the intrinsic order is ultimately dependent on choice is irrelevant, because some order must be chosen for, or else the system would be meaningless. — Metaphysician Undercover

No, you've simply shown how external order has been switched for internal order. And now I've shown the problem which arises from this switch, the contradictory, therefore impossible "empty set", which makes the inclusion of zero an inconsistency. — Metaphysician Undercover

As I say, the idea that you've gotten rid of the order properties is just an illusion. The order inheres within each individual number, as the definition of that specific set. Rather than simply being an external property of a number, as an object, and how it relates to other numbers, order is now an internal property of the number itself, as a set.. — Metaphysician Undercover

I argue the exact opposite, that you are consistently wrong about this. It is exactly "two copies", just like the word "same" here, and the word "same" here, are two distinct copies, even though we say it's the same word. Look, we are talking the meaning of symbols here. "A=B" means that that symbol A has the same meaning as B, it does not mean that A signifies the same entity as B, without additional information. However, the additional information in this case indicates that what is signified by A and B is a set, "the same set". But a set is not a thing, it is a group of things, grouped by a categorization such as type. Therefore this is an instance of "the same meaning", signified by A and B (indicated by "type"), not an instance of the same entity signified by A and B. This is just like when we use the same word twice when the word has meaning, rather than referencing a particular object. We say that the word has the same meaning, just like we might say A and B have the same meaning, in your example. — Metaphysician Undercover

Exactly, and this is a different meaning of "same" from the meaning of "same" in the law of identity. That is the point. In the law of identity "same" means a lot more than simply having the same members (what I called a qualified "same"), it means to be the same in every possible way ("same" in an absolute, unqualified way), — Metaphysician Undercover

I totally agree with that, that's what "same" means in this context. — Metaphysician Undercover

The problem is that it does not mean what you stated above: "They are in fact the identical set, of which there is only one instance in the entire universe". The set is an imaginary thing, indicated by meaning, it is not something in the universe. So it's not even coherent to say that there is one instance of that set, it's not even a thing which has an instance of existence, it's just the meaning of a symbol. So you speak of "the same set", and claim there is only one instance of that set, but this would be taking a different meaning of "same", which refers to instantiated things, and applying it to "same set", which really means having the same meaning, and not referring to one instantiated thing. Do you see the difference between referring to one and the same thing with a name, "MU", and using a word which has meaning, like "person", without any particular thing referred to? Person refers to a type, so it has meaning, just like "set" refers to a type, so it has meaning. These do not refer to instantiated things of which we could say there is one instance of, they refer to ideas. — Metaphysician Undercover

You have a hidden element here, known as freedom of choice. The "multiple possible outcomes" are only the result of this hidden premise, you have freedom to choose. That premise overrules "the rules of the game", such that the two are inconsistent. In other words, by allowing freedom of choice, you allow for something which is not "consistent with the rules of the game", this is something outside the rules, the capacity to choose without rules. — Metaphysician Undercover

In that thread they state that "any set of sentences can be a set of axioms." I want to distinguish between what is (i.e. actual) and what can be (i.e. potential). It is tempting to actualize everything and declare that there are uncountably many mathematical truths. However, I would argue that these truths are contingent on a computer constructing them. When I speak of finite necessary truths I'm referring to the rules of logic itself. — keystone

I'm trying to establish a view of calculus which is founded on principles that are restricted to computability (i.e. absent of actual infinities). You don't have to abandon your view of actual infinities to entertain a more restricted view. Perhaps we can set aside the more philosophical topics and return to the beef. — keystone

The term 'line' comes loaded with meaning so to start with a clean slate I'll use 'k-line' to refer to objects of continuous breadthless length (in the spirit of Euclid). I'll use <a,b> to denote the k-line between a and b excluding ends and <<a,b>> to denote the k-line between a and b including ends. If b=a, then <<a,b>> corresponds to a degenerate k-line, which I'll call a k-point and often abbreviate <<a,a>> as "a". I'll call the notation <a,b> and <<a,b>> k-intervals. — keystone

The systems always start with a single k-line described by a single k-interval (e.g. <-∞,+∞>). A computer can choose to cut the k-line arbitrarily many time to actualize k-points. For example, after one cut at 42, the new system becomes <-∞,42> U 42 U <42,+∞>.

The order relation comes from the infinite complete trees.

Are we at a place where we can we move forward? — keystone

So I translate all talk of the lamp into abstract structure in which "0, 1, 0, 1, ..." is aligned with "1, 1/2, 1/4, ...". — Ludwig V

I agree. But I have some other problems about this. I'll have to come back to this later. Sorry. — Ludwig V

Possible outcomes can indeed be inconsistent with each other. But if they are inconsistent with each other, they can't both be actual at the same time. You can't drive down the road and turn left and right at the same time. — Ludwig V

Before we even consider if and when we push the button it is established that the lamp can only ever be on if the button is pushed when the lamp is off to turn it on. — Michael

Everybody who has been around dementia patients will see what is going on. The patient's regress to a child-like state is symptomatic of dementia: — Lionino

The 70 year old anti-vax conspiracy theorist who has dealt with literal brain worms... we really have a great slate of candidates this year. — Mr Bee

These are our premises before we even consider if and when we push the button:

P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 10:00 — Michael

For set theory, an example is Hindman's 'Fundamentals Of Mathematical Logic' — TonesInDeepFreeze

The Wikipedia article you mentioned is not well written. (1) It doesn't give an axiomatization and (2) It doesn't mention that we can have other symbols in the signature and that by a schema we can generalize beyond a signature with only '='. — TonesInDeepFreeze

So Ax x=x is also an axiom incorporated into set theory. — TonesInDeepFreeze

And for set theory, his 'Elements Of Set Theory'. — TonesInDeepFreeze

I still reckon there's a possiblity that if a new nominee appeared at the eleventh hour, there could be a huge rush to them, just on account of him/her (probably 'him') being an alternative to the godawful mess that now exists. — Wayfarer

You might think so from what I said, but he was young and pretty enthusiastic about teaching the subject. We had numerous worksheets that eventually led to the construction of the exponential function. So, his comment at the end came as a bit of a surprise. — jgill

We'll see. — Wayfarer

I presume as the President that he's is subject to regular medical examinations, right? And that if he were displaying symptoms of senile dementia, this is something that these examinations would detect? — Wayfarer

And that, were it detected, the responsible medical officers would report it and not try to conceal it? — Wayfarer

Ronny someone. Got that reward for being a compliant flunky and saying good things about the Orange Emperor. That'll guarantee you a place in the MAGA pantheon. — Wayfarer

Rather than sorting out your questions in this disparate manner, it would be better - a lot easier - to share a common reference such as one of the widely used textbooks in mathematical logic. I think Enderton's 'A Mathematical Introduction To Logic' is as good as can be found. And for set theory, his 'Elements Of Set Theory'. — TonesInDeepFreeze

I said that classical mathematics has the law of identity as an axiom and that classical mathematics abides by the law of identity. — TonesInDeepFreeze

I addressed that. Written up in another way:

Ordinarily, set theory is formulated with first order logic with identity (aka 'identity theory') in which '=' is primitive not defined, and the only other primitive is 'e' ("is a member of"). — TonesInDeepFreeze

But we can take a different approach in which we don't assume identity theory but instead define '='. I don't see that approach taken often. — TonesInDeepFreeze

But both approaches are equivalent in the sense that they result in the exact same set of theorems written with '=' and'e'. — TonesInDeepFreeze

No, I am not saying any such thing. — TonesInDeepFreeze

(1) I don't think I used the locution 'logical identity'.

But maybe 'logical identity' means the law of identity and Leibniz's two principles.

Classical mathematics adheres to the law of identity and Leibniz's two principles. — TonesInDeepFreeze

The identity relation on a universe U is {<x x> | x e U}. Put informally, it's {<x y> | x is y}, which is {<x y> | x is identical with y}.

Identity theory (first order) is axiomatized:

Axiom:

Ax x = x (law of identity)

Axiom schema (I'm leaving out some technical details):

For any formula P(x):

Axy((P(x) & x = y) -> P(y)) (Leibniz's indiscernibility of identicals) — TonesInDeepFreeze

But identity theory, merely syntactically, can't require that '=' be interpreted as standing for the identity relation on the universe as opposed to standing for some other equivalence relation on the universe. — TonesInDeepFreeze

And Leibniz's identity of indiscernibles cannot be captured in first order unless there are only finitely many predicate symbols. — TonesInDeepFreeze

So we make the standard semantics for idenity theory require that '=' does stand for the identity relation. And then (I think this is correct:) the identity of indiscernibles holds as follows: Suppose members of the universe x and y agree on all predicates. Then they agree on the predicate '=', but then they are identical. — TonesInDeepFreeze

(2) The axiom of extensionality is a non-logical axiom, as it is true in some models for the language and false in other models for the language. — TonesInDeepFreeze

As mentioned, suppose we have identity theory. Then we add the axiom of extensionality. Then we still have all the theorems of identity theory and the standard semantics that interprets '=' as standing for the identity relation, but with axiom of extensionality, we have more theorems. The axiom of extensionality does not contradict identity theory and identity theory is still adhered to. All the axiom of extensionality does is add that a sufficient condition for x being identical with y is that x and y have the same members. That is not a logical statement, since it is not true for all interepretations of the language. Most saliently, the axiom of extensionality is false when there are at least two urelements in the domain. — TonesInDeepFreeze

In sum: Set theory adopts identity theory and the standard semantics for identity theory, and also the axiom of extensionality. With that, we get the theorem:

Axy(x = y <-> Az((z e x <-> z e y) & (x e z <-> y e z)))

And semantically we get that '=' stands for the identity relation. — TonesInDeepFreeze

And if we didn't base on identity theory, then we would have the above not as a non-definitional theorem but as a definition (definitional axiom) for '='; and we would still stipulate that we have use the standard semantics for '='. — TonesInDeepFreeze

Think of it this way: No matter what theory we have, if it is is built on identity theory, then the law of identity holds for that theory, and that applies to set theory in particular. But set theory, with its axiom of extensionality, has an additional requirement so that set theory is true only in models where having the same members is a sufficient condition for identity. — TonesInDeepFreeze

No Cantor crank would ever have the self-awareness to know that he or she is a crank. — TonesInDeepFreeze

Here are quotes from my earlier posts. You don't have to read all bullets as they all say the same thing. I'm just trying to highlight that the confusion is not for lack of me trying.

"Suppose I introduce a new concept called 'k-interval' to define the set of ANY objects located between an upper and lower boundary. Would you then consider allowing objects other than points into the set?"
"Yes, the endpoints are rational, and the object between any pair of endpoints is simply a line."
"Revisiting the analogy above, when I utilize an interval to describe a range, I am referring to the underlying and singular continuous line between the endpoints"
"Yet, between each tick mark, there exists a bundle of 2ℵ0 points to which we can assign an interval."
"I propose that we redefine the term interval from describing the points that lie between endpoints to describing the line that lies between endpoints."
"One thing I need to make clear though is that when I write (0,4) I'm referring to a bundle between point 0 and point 4. I'm not referring to 2ℵ0 points each having a number associated with them. " — keystone