Right. Thanks.

Godel-Rosser is:

If T is a consistent formal theory adequate for arithmetic, then T is incomplete.

The steps in the proof depend on T being a formal theory.

What would "extend the import to non-formal languages" mean?

The crank — Lionino

Hey, calling cranks 'the crank' is my schtick. Please don't steal my act!

1. If something is possibly necessary, is it necessary?

Under S5 (one type of modal logic), the answer is "yes". — Michael

I'm very rusty in modal logic. How do you derive ('n' for necessary, 'p' for possible):

pnQ -> nQ

/

We start with:

Df. pQ <-> ~n~Q
therefore, nQ <-> ~p~Q

Ax. n(Q -> R) -> (nQ -> nR)

Ax. nQ -> Q

Ax. pQ -> npQ

And at least one easy theorem:

Th. Q -> pQ

How do you derive:

pnQ -> nQ


This is how far I get:

Suppose pnQ

Show nQ (or show ~p~Q)

Suppose ~nQ (or suppose p~Q) to derive a contradiction

?

We can state the indiscernibility of identicals as a first order schema, no matter how many nonlogical symbols there are in the language.

And we can state the identity of indiscernibles as a first order schema if there are only finitely many nonlogical symbols in the language.

But it's interesting that we cannot state the identity of indiscernibles as a first order schema if there are infinitely many nonlogical symbols in the language.

wut? axiom of infinity. what's wrong with you tonight? — fishfry

The axiom of infinity is how we prove that there is a set that has every natural number as a member.

From the axiom of infinity, we derive that there is a unique least inductive set.

Then we define: w = the least inductive set.

Then we prove that all the natural numbers are members of w.

My point is that we have to be careful in thinking of making this definition:

w = the limit of the sequence of all the natural numbers

since the domain of that sequence is the set of natural numbers, which would already have to have been defined, and so we would have already defined w.

w = the ordinal limit of the sequence of all the natural numbers*

is a theorem, but it would be tricky were it a definition.

* Given a reasonable definition of 'limit of sequence of ordinals'.

/

This is yet another instance of you lashing out against something that I wrote without even giving it a moment of thought, let alone maybe to ask me to explain it more. Your Pavlovian instinct is to lash out at things that you've merely glanced upon without stopping to think that, hey, the other guy might not actually being saying the ridiculous thing you think he's saying. Instead, here you jump to the conclusion that "there's something wrong" with him.

I need to go from x = y to saying that for all z, x in x iff z in y. — fishfry

A typo there? I think you meant 'z in x iff z in y'?

There's a point from a while back. Maybe we can fix it.

I said that

ExAy y e x

is consistent.

You disputed that.

So I pointed out that I am not saying it is consistent with set theory, which has the axiom schema of separation from which we derive:

~ExAy yex.

Rather, it is consistent just as it stands alone.

I said "It is consistent onto itself." Yet, you still disputed me. Much later it dawned on me that you were thinking that I meant 'onto' as with a surjection. But I meant 'onto itself' to mean 'in and of itself'. And later I found out that people don't usually say 'onto itself' that way. So I saw that I had lapsed in English.

So here we are, and I am hoping that you see that I was correct that

ExAy yex

is consistent in and of itself, even though not consistent with the axiom schema of separation.

At the time I proved by adducing this model:

U = {0}

'e' stands for {<0 0>}

The converse of the axiom of extensionality where he says that the converse "follows from the substitution property of equality." — fishfry

Exactly. That goes right with what I've been saying.

Without sarcasm I say that it gives me a good feeling that reason, intellectual curiosity and communication have won the day finally.

I know you're kidding. But underneath there lies an actual point for me, which is that I don't think you know how insulting you are in certain threads when you read (if it can be called 'reading') roughshod over my posts, receiving them merely as impressions as to what I've said, so that you so often end up completely confusing what I've said and then projecting your own confusions onto me.

But I do appreciate that you quoted Cole Porter's so charming and magical lyric. And there was another special musical moment for me today, so my evening was graced.

enderton page ref please or st*u. second time i'm calling your bluff on references to your magic identity theory. — fishfry

Did you mean for that to be in the 'Infinity' thread?

In that thread, you've now seen that I already had given you the Enderton pages yesterday and I gave them to you even though you had not asked for them. There's no bluff and never has been. I've been giving you post after post of correct corrections, information and explanations. It's not my fault that you regard that as inimical.

That might be. I'm speaking in broad terms about them in that regard. If the article draws a needed distinction then I should say that they are at least akin. — TonesInDeepFreeze

Nope. I am consistent with the SEP article. The context in this discussion is plain predicate logic where substitution works, not intensional contexts.

They did say that "Leibniz’s Law must be clearly distinguished from the substitutivity principle ..." so perhaps that's pushback to your claim. — fishfry

That might be. I'm speaking in broad terms about them in that regard. If the article draws a needed distinction then I should say that they are at least akin.

But there are actually two principles — fishfry

Right. I discussed that in about my first post on the subject in this thread.

So in set theory Ax x=x is redundant.
— TonesInDeepFreeze

in which case you agree with my main point and there is nothing more to say. — fishfry

A few posts ago, I explained exactly why there is more to say.

Again:

Whatever was your main point, one of your points, and the one we've been thrashing over for dozens of posts, is that we only need the axiom of extensionality for identity. And I've explained and explained for you, a trillion ways to Sunday, exactly why that is not true.

So I appreciate that you are now writing much shorter posts — fishfry

I write long posts because you post so many incorrect claims and confusions about the subject and confusions about what I've posted, and, often enough, I give meaty explanations.

I can't keep up. — fishfry

I am hopelessly behind composing posts in at least a few threads. Even years behind in threads that I just had to let go because I really should be spending my time on other things more important than posting.

If you mean that it would help for my posts to link to yours, then I'll hope not to forget doing that each time.

My preference regarding you is that you don't gloss my posts and jump to conclusions that I've said something I didn't say but that you think I must have said in you own confusions or lack of familiarity with the concepts or terminology.

can't find anything at all on page 83 — fishfry

That is where he gives the semantics for '=', as I mentioned that '=' is given a fixed interpretation.

So I saw a ref to equality on 112 of enderton that had nothing to do with set theory — fishfry

Yes, I said that it doesn't mention set theory, but rather it is a place to see the logical axiom schema for first order logic with equality.

However, as you have agreed, set theory uses first order logic with equality. So there you have it.

And, yes, at least twice, I said myself that Ax x=x can be derived from the axiom of extensionality. So in set theory Ax x=x is redundant. But the axiom schema for the indiscernibility of identicals is not redundant in set theory.

This is my point. — fishfry

But that is just a part of the picture. And you keep slapping about saying that we need only the axiom of extensionality and don't need the indiscernibility of identicals. And when I pointed out we can't prove

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

from the set theory axioms without the indiscernibly of identicals, you claimed that you easily can.

So, again, I say that I'd love to see your attempt.

And you can look at the SEP article 'Identity' where you'll see:

Leibniz’s Law, the principle of the indiscernibility of identicals, that if x is identical with y then everything true of x is true of y.

I hope you know that 'the crank' does not refer to you. If that was not clear in the context, then I should have made it clear.

I already gave you the pages:

The Enderton reference was to the identity axioms. See page 112 in the logic book. And also, on page 83, he specifies satisfaction regarding '=' so that it adheres to interpreting '=' as the identity relation. — TonesInDeepFreeze

And I gave you page number and line numbers for Shoenfield:

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

I said that the indiscernibility of identicals is formalized in identity theory. I didn't say that any particular formalization mentions it with the phrase 'the indiscernibility of identicals'. The principle was enunciated by Leibniz. But in mathematics, it's often called 'the principle of substitution of equals for equals'. And in modern logic, it is an axiom schema in the manner I've posted, which is equivalent (though notation and details differ) to Enderton and Shoenfield, for example.

I'm giving you a lot of the same information and explanation over and over, since you skip over it over and over.

a trip to the moon on gossamer wings — fishfry

Seeing just that one phrase from the great song made my night. Such a soul satisfyingly beautiful song by a gigantically great composer.

Only countably many interpretations of each sentence. — fishfry

I'm talking about interpretations for languages as discussed in mathematical logic.

There are uncountably many sets, so there are uncountably many universes for interpretations.

Or, another way: Consider just one uncountable universe. Let the language have at least one individual constant. Then there are uncountably interpretations as each one maps the constant to a different member of the universe.

I don't propound the notion that that approach could be adapted for natural languages too, but it doesn't seem unreasonable to me.

I'll ask again:

But if you start from that there is no bijection, and then prove it by:
If there is a bijection then there is a surjection
There is no surjection.
Therefore, there is no bijection.

Isn't that a proof by contradiction?
— ssu — ssu

I gave you a very detailed answer. I can't do better than what I already wrote. Or, if you like, let me know what you don't understand in my post.

Cantor's diagonal argument says that any list of reals is incomplete. We can prove it directly by showing that any list of reals (not an assumed complete list, just any arbitary list) is necessarily missing the antidiagonal. Therefore there is no list of all the reals. — fishfry

Exactly.

I don't want to watch a video right now.

theories (interpretations) — fishfry

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.

w can be defined such that it is the limit of the sequence of the natural numbers. — fishfry

Of course, w is a limit ordinal, and it is the ordinal limit of the sequence of all the natural numbers.

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.

The head sophist at TPF, TIDF, continues to defend sophistry by arguing that intelligible objects are consistent with the law of identity.
— Metaphysician Undercover

Drat those sophists. Are they in the room with us right now?

Oh I see. Tones. — fishfry

That exchange deserves nothing more than a snort.

The crank still can't vindicate his claims sets by answering what is the inherent order of the set whose member are the bandmates in the Beatles.

And fishfry is lately asking me to give examples regarding identity when I've given three of them three times already, while he has not shown a proof of the transitivity of equality without using sub of equals for equals, though he claims he can do it.

Too late to wriggle out — fishfry

No wiggling. It was faulty of me to reference that book without specifying that I do not claim it discusses the identity axioms.

Ah, you have 'fessed up after all. — fishfry

I had previously admitted that I would have been mistaken if I referenced the book regarding identity theory.

So, no, not "after all".

You are making more of this than I intended for you to make. — fishfry

You could have stopped the first time I recognized my lapse.

And I make of it what it is worth: Being clear as to what was actually posted.

I'm mocking you for saying that you agreed with a point I made — fishfry

I know that. And your mocking is sophomoric. There is nothing amiss in agreeing on a point with someone but commenting that nonetheless their reasoning about it is poor.

If you cared more about the subject at hand then in prevailing with lame smart-aleckisms, then you could go back to the post to see my substantive point.

I have a bad habit of tweaking and needling people — fishfry

You're not good at it. And I don't buy that your motive is just to josh but not also imbued with putdown as a kind of trump card.

I don't know "nothing" about the matter. I know logic as it's used in math, but did not study enough formal predicate logic. Indiscernibility of identicals I know of in other contexts, and am genuinely surprised to hear that it's incorporated into set theory. — fishfry

You don't know enough to know that when we use the principle of substitution of equals for equals in mathematics, including set theory, we are in effect using the principle of the indiscernibility of identicals, whether we explicitly recognize it or not. And, with first order logic, which codifies and formalizes the reasoning for classical mathematics, we do explicitly formulate the principle in an axiom schema.

As I recall, the reason I mentioned the subject lately, and with that fancy name, is that the law of identity had been mentioned as historical and fundamental. My point was that also the indiscernibility of identicals is historical and fundamental. Indeed, with those two historical ideas, we axiomatize first order identity theory.

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

It's interpreted as the axiom of extensionality in set theory. Which doesn't actually require identity, and I've asked for a specific example to prove otherwise. — fishfry

I have to repeat myself:

(1) Interpretation is semantical. The axiom of extensionality is syntactical.

(2) Even just syntactically, the axiom of extensionality is not a definition, in the sense of a syntactical definition.

(3) If set theory didn't have the identity axioms, then, even with the axiom of extensionality, set theory would wouldn't even get very far off the ground.

(4) I already gave you specific examples about three times!

Without the indiscernabilty of identicals you can't prove:

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

x = y -> y = x

(n is even & j = n) -> j is even

If I have two sets, and I want to know if they're equal, I apply extensionality. Not identity. — fishfry

I've said, extensionality gives a sufficient condition for equality, but not a necessary condition for equality.

Yes, to prove x = y, then it suffices to prove Ax(z e x<-> z e y)

But that's not the only thing we do with '='.

We also use equal to reason this way:

x has property P, and y = x, so why has property P.

And that uses the indiscernibilithy of identicals.

Sometimes it's called "substitution of equals for equals".

When a school kid says:

"2 = 1+1, so 1/(1+1) = 1/2"

that is using substitution of equals for equals.

So, to codify and formalize use of subtsiution of equals for equals, we make it an axiom. And that axiom is a fomalization of the identity of indiscernibles,

I ask for a clear clear clear clear clear refutation or counterexample. — fishfry

I gave three, three times!

But you claimed that you could prove:

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

using just the axiom of extensionality (or, even all the set theory axioms but not identity theory axioms), and I said:

THEN DO IT.

C1 is a premise.
— TonesInDeepFreeze

It’s not, it’s a valid inference from the premises. — Michael

The premises don't not specify that the button is ever pushed.

The premises do not specify that there are only two states, unless, in this very hypothetical context we are clear that 'Off' is defined as 'not On', though it does seem reasonable that that is implicit.

rantings — fishfry

Making clear corrections, giving generous explanations, commenting the deplorable methods of cranks, and posting ideas in general is not ranting.

/

Looking back:

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

So I see now that I recommended Enderton's set theory book in general. I didn't say that it is specifically a reference to the fact that set theory is based on identity theory (first order logic with equality).

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.

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, and to that extent, my recommendation was faulty.

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.

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.

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

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

(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

Ok — fishfry

/

You have MUCH BETTER REASONS than I do. Ok. — fishfry

In that instance, yes, and that is made clear by what I wrote. But rather than address the substance of the matter, you opt for ill-premised sarcasm about the exchange.

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

I admitted to being a logic slacker. — fishfry

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.

I no longer know what we were talking about. — fishfry

We were talking about how '=' is interpreted.

Mostly, I would be very interested to see your proof of:

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

You may use only the law of identity Ax x = x and the axioms of set theory. You may not use substitution of equals for equals, unless you prove it from only the law of identity and the axioms of set theory.

For reference, here's the axiom of extensionality:

Axy(Av(v e x <-> v e y) -> x = y)

Here's a head start to a dead end:

Assume x = y & y = z

Show x = z

To show x = z it suffices to show

Av(v e x <-> v e z)

Go to town on it! I'd like to see what you got!

/

Or try proving:

x = y -> y = x

Of course it is. It's an axiom. It says what is true about all the things we call sets. Therefore we can characterize the world of things into sets and non-sets, according to whether they satisfy the axiom. So axioms serve as definitions and vice versa. They are the same thing. — fishfry

I addressed that in detail. You could reread what I wrote.

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

That's fine, but that's one of the points where you lose me. Why do you care, or why do you think your doing so will make me understand something I didn't understand before? — fishfry

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.

I've never heard of identity theory except in the context of many of the Wiki disambiguations. And when I showed you the most likely meaning, you rejected it. So I have no idea what identity theory is. — fishfry

We're going around full circle.

(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!

(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).

(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.

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

I don't recall even having an opinion about this, let alone expressing it in this thread. — fishfry

You had written.

But if set theory adds an axiom, then clearly it is not the same thing. It's something else, a new thing. — fishfry

And I agreed with that, but wished to phrase it as I prefer.

I said reasons and you said better reasons? Ok. Your reasons are much better than my reasons for believing things we both agree on. — fishfry

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

Every attempt I make to understand you is wrong. So maybe just give up because I don't get it. — fishfry

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

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

Not really. — fishfry

You just say "not really" without basis and without even taking a moment to reflect on it.

Again, without the indiscernibility of identicals we wouldn't be able to prove the theorems regarding identity that are required for our formalized mathematics.

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

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

Of course we can, straight from the axiom. — fishfry

Then you could show it. Give the proof steps, using only the axiom of extensionality and the law of identity (and even also any number of the other set theory axioms). This is the second time I've suggested that you show what you think is a proof:

And this is the second time I say:

WARNING: You may not use substitution of equals for equals unless you prove that principle from only the law of identity and the set theory axioms.

And now I'll add:

HINT: You won't be able to prove the principle of substitution of equals for equals, since it is itself tantamount to the indiscernibilithy of identicals.

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.

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

The identity relation is an equivalence relation. — fishfry

Of course it is. My remarks imply that I know that it is. My point though is that it is a particular equivalence relation, which is that x is identical with y iff x is y and not merely that they <x y> is in an equivalence relation.

EVERY idea I toss out to try to relate to what you're saying, you reject. — fishfry

'EVERY' (all caps no less) is an overgeneralization. It is proven wrong even in my previous post where I agreed with you that set theory is more than identity theory.

But, yes, I do comment often where I find errors.

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

Irrelevant to anything I can relate to, in this conversation or in general. — fishfry

Relate to what you will, but the statement I made corrects your false claim that that we do not need the indiscernibility of identicals for doing even ordinary math, even if it is a reasoning principle that mathematicians don't bother to log in their arguments. It is, as they say, "the water the fish lives in". The substitution of equals for equals is ubiquitous.

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

I never stipulated to it. — fishfry

So what? In logic it is ordinarily stipulated.

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.

EDIT LATER: Disregard this post. I hope to post a revision.

I'll try this:

Suppose:

There are two states F and N.

At any moment either F is active or N is active and not both.

For every natural number n>0, there is a time T(1/(2^n)).

At time T(0), F is the active state.

For every natural number n>0, the active state changes at time T(1/(2^n)).

The active state changes only at time (T(1/(2^n)) for some natural number n.

Question: What is the active state at time T(1)?

I have no idea what "the lamp" is. — fishfry

It doesn't matter to me what the lamp is.

I can regard the problem abstractly, in terms just of:

time
two states
sufficient and necessary conditions for changing states

Mentioning a lamp and it being on or off and a button that is pushed are, for me, all just visualization aids that are dispensable. 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.

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

(1) Why not use 11:00 rather than 10:00? Usually the problem concerns 11:00 to 12:00, which is tidy for the halvings of the durations. (I'll use 11:00.)

(2) We could do without pushing a button and even the lamp. We could couch it in more abstract terms.

(3) Your premises have only one necessary condition for the lamp changing state, which is P1.

This is consistent with your premises:

The lamp is off at 11:00. The button is pushed at 12:00 and the lamp goes on.

So I think you've left out a lot of what you need in your premises.

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.

Who was the famous logician?

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.

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. 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.

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

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

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

That's not right.

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.

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

— TonesInDeepFreeze

Ok at this point, I am wondering: Why are you telling me this? I don't understand what you want me to know about this. What problem are we trying to solve? — fishfry

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.

I am certain I never said we don't need identity! — fishfry

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.

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, which would comport with your view that the axiom of extensionality is a definition. So, I mentioned that, for example, from the axiom of extensionality alone we cannot prove:

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

I'm all for the law of identity. A thing is equal to itself. That's good do know. In fact it helps make equality an equivalence relation with exactly one item per equivalence class. — fishfry

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

(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.

Moreover, we want to ensure that '=' stands not just for an equivalence relation but for, indeed, the identity relation. And to do that we have to make the semantic stipulation that '=' is interpreted as standing for the identity relation on the universe.

Alas. — fishfry

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.

I have no idea what pickle you see.
— TonesInDeepFreeze

Then my attempt to explain my take on the subject we're discussing failed. — fishfry

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

But I've not had any pickle in that way and I have not read about such a pickle. So I can know what pickle you have in mind only if you tell me.

If that is the price of conversing further with you on these matters — fishfry

Of course it is not.

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.

I will agree with you that identity is implicitly in extensionality, in the sense that two sets are equal if they have the "same" elements. — fishfry

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

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.

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):

Axy((P(x) & x=y) -> P(y) (if x is y, then whatever holds of x then holds of y, i.e. "the indiscernibility of identicals")

Then add the axiom of extensionality:

Axy(Az(z e x <-> z e y) -> x = y) (x and y having the same members is a sufficient condition for x and y being identical)

We need identity to know when two elements are the same. — fishfry

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

x = y <-> y = x

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

(P(x) & x = y) -> P(y) (for example, (x is finite & x = y) -> y is finite)

etc.

/

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.

/

Now, since the elements of sets are other sets (barring urelements for the moment), I can see that there's a bit of a pickle. I''m not sure how this pickle is resolved. — fishfry

I have no idea what pickle you see.

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

Is it? — fishfry

No.

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 proofs of invalidity by counterexample)

Then:

A Mathematical Introduction To Logic - Enderton

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.

first is assumed that all reals, lets say on the range, (0 to 1) can be listed — ssu

If that is considered a form of reductio ad absurdum, then every proof of a negation is proof by a form of reductio ad absurdum.

In a natural deduction system, the way to prove a negation ~P is to assume P, derive a contradiction, and infer ~P.

In an ordinary Hilbert system, the way to prove a negation ~P is to prove, for some Q, P -> Q and ~Q, and infer ~P.

Yes, those are like "cousins" of one another. And they can be derived from one another as derived rules in the systems.

But again, if using modus tollens is considered a form of reductio ad absurdum, then any proof of a negation is a form of reductio ad absurdum.

Note that both of those are intuitionistically valid. What are not intutionistically valid are:

Assume ~P, derive a contradiction, and infer P.

~P -> Q and ~Q, and infer P.

/

Also there are different terminologies:

reductio ad absurdum

indirect proof

proof by contradiction

So we need to be clear whether the intuitionistically valid form or the intuitionistically invalid form or both are referenced.

/

You mentioned 'indirect proof' and you said:

first is assumed that all reals, lets say on the range, (0 to 1) can be listed — ssu

My point was that we do not need to assume that all the reals are listed. "All the reals are listed" would be P in the remarks above.

Now you've switched to pointing out that modus tollens is used.

And this has nothing to do with anti-diagonalization.

It's garden variety modus tollens:

If there is a bijection then there is a surjection
There is no surjection.
Therefore, there is no bijection.

No need for a reductio assumption.