If it is consistent, it is not complete; if it is complete, it is inconsistent. — Agent Smith

If it is recursively axiomatized and has enough of arithmetic, then if it is consistent then it is incomplete (which is to say that if it is complete then it is inconsistent).

Many years ago I was watching the Academy Awards (don't ask me why I would waste my time that way). Whoopi Goldberg was at the microphone, and she said something like, "We don't need the space program. We should get rid of it. Nothing ever came out of the space program", as she was being broadcast via satellite. You gotta love it.

we could play around with the axioms to disallow irrationals, oui? — Agent Smith

Si.

I've told you about a million times already, you can have axioms for whatever you want*, even inconsistency if that's your thing.

* 'ceptin sometimes you can't have all of what you want at the same time, as we found out from Godel.

My favorites are cranks who say that mathematical logic, even just sentential logic, is all wrong, as they are typing and reading on computers that are packed to the panels with Boolean programming and would not even exist if not for the invention of modern digital computers in the very crucible of mathematical logic.

Communication is bad because 'communication' is from the same root as 'communism', which is bad because communism is Marxism, which is bad because Richard Marx sang lousy songs.

Couldn't there be something wrong with both? — Agent Smith

Couldn't there be something wrong with you?

Mathematicians come up with general formulas, involving pi and other irrational numbers. Isn't it the case that engineers make use of those general formulas, from which they can decide what specific specific values to use as close enough for the task at hand?

If that is the case, then it's a mistaken premise that we can decide on a greatest practical number, or greatest number of places to approximate pi. Science and engineering doesn't work with just specific numbers but also rather with general formulas. I take engineers at their word that they use trigonometry with pi all over the place. Not that they plod along with just individual values.

Stick with your earlier theory that irrational numbers are bad because 'irrational' also means illogical. That one is a doozy.

Now that you've explained some of your terminology (I had no way of knowing that, out of the blue, you would be using category theory), I'm going to revise some of my earlier comments:

(1) Your bit about choice didn't make sense to me earlier, but I might have been incorrect to mention it in connection with proving that N is Dedekind infinite.

(2) I said your remarks about N were not a definition. Perhaps they are in a context of category theory.

(3) Ordinarily I take 'intensionality' to be about intensional contexts such a modal, epistemic, etc. operators. Now that you've clarified your sense of 'intensionality' and 'extensionality', I don't dispute that, in that sense, the ordinary definitions regarding infinite sets are not extensional. But I don't like that terminology because it jams against the ordinary sense of 'extensionality' we use in set theory.

(4) Perhaps you're not down the Rabbit Hole. But I'm cautious not to fall into it.

/

On the other hand, only by extraordinary shifting of context would you not be incorrect in your description of the hotel as finite but always expanding. Hilbert meant an infinite hotel, not a finite one always expanding. If you want to set up a different story from Hilbert's then of course that is fine. But we shouldn't conflate them.

EDIT NOTE: I meant to include 'not' before 'down' in the comment about the Rabbit Hole. It's corrected now.

You seem to be either (1) Using your own made up ways of talking about this or (2) Speaking in a different context from me.

If we don't agree on how we speak about the subject or our context for the way we speak about the subject, then we can't communicate.

My context is ordinary mathematical logic, which is the context of Hilbert himself. Though his Hotel predates the fuller context of mathematical logic, at least we're in the ballpark.

If you have a different context in mind, then one needs to know where to can reference it.

So, from my context, here goes:

extensional meaning — sime

I don't look for sets to have a meaning. Word, symbols, sentences have meanings. Where do authors speak of sets having meaning?

the items it refers to — sime

Sets don't refer to items. Sets have members. Where do authors speak of sets referring to items?

definition of the set in terms of a formula — sime

Granted, authors often speak of 'defining sets'. But to be accurate, there are two senses of defintion:

(1) Syntactical. Defining a symbol.

Example

'N' is the symbol.

The definition of 'N' is

N = x <-> (x is an inductive successor set & x is a subset of every inductive successor set)

Sometimes, we have definitions such as this:

B = {0 5 8}

I take that is an example of what you mean by an 'extensional definition'.

(2) Semantical. For a given interpretation of a language, showing a formula using n free variables that is satisfied by certain n-tuples of members of the domain of the interpretation. (That's putting it roughly, though.)

Example (with first order PA)

With the standard model

x = x

defines the set of natural numbers.

category — sime

I didn't know you're using category theory.

the presence of Choice causes all infinite sets in ZF to become Dedekind infinite by default — sime

Of course, though I wouldn't state that way.

which is a major failing of ZFC in ruling out the only sort of "infinite" sets that have any pretence of physical realisability. — sime

It rules out that there are infinite sets that are not Dedekind infinite. But I don't know your definition of 'physical realizability'. Sets are not physical anyway (though, outside of set theory, we could have sets whose members are physical objects). Also, even without choice, we can't prove that there do exist infinite sets that are not Dedekind infinite.

There's an interesting question: In ZF we cannot prove 'There does not exist an infinite set that is not Dedekind infinite'. But do we ever have this situation?:

A definition of 'Q' and a proof of 'Q is infinite' but such that there is not also a proof of 'Q is Dedekind infinite'.

In other words, it's conceivable that even though we can't prove that every infinite set is Dedekind infinite, for any infinite set we define, we can prove that it is Dedekind infinite. Hmm, not sure, but I think that might be the case, since a definable set will be constructible (?) and the axiom of choice holds for constructible sets. (?)

extensionally meaningful — sime

Definition of 'extensionally meaningful'?

And, if you respond with word salad, as you are wont to do, then I can't help you.

So if one writes down an inductive definition of the natural numbers

1 + N <--> N

where <--> is defined to be an isomorphism — sime

That's not a definition of anything, let alone the set of natural numbers.

then to say N is "Dedekind-Infinite" means nothing more than to restate that definition. — sime

'N is Dedekind infinite' means that there is a 1-1 correspondence between N and a proper subset of N. There's no need to drag isomorphism into it. The function {<j j+1> | j in N} is provably a 1-1 correspondence between N and a proper subset of N, so it proves that N is Dedekind infinite (and notice, contrary to your incorrect claim, choice is not involved). But that proof is not a definition of anything, let alone of the set of natural numbers.

And intensionality has nothing to do with this.

Your messages each time are as good as scribbled postcards from the Rabbit Hole.

pi is not a ratio of two rational numbers.

pi is the ratio of the circumference of any circle and its diameter. But if the diameter is rational then the circumference is not. So still pi is not the ratio of two rational numbers.

So there is not a contradiction.

I don't know what "problems" in mathematics are supposed to have been caused by pi.

https://www.jpl.nasa.gov/news/on-pi-day-how-scientists-use-this-number

So I guess people that don't like pi being used in mathematics have not need for such things as satellite technology, etc. Fair enough.

But we have to distinguish between the extensional concept of a number of hotel rooms that can be built, visited, observed, realized etc, versus the intensional concept of a countably infinite set of rooms. — sime

There's no consideration of intensionality in the illustration.

a piece of syntax representing an inductive definition of the natural numbers. — sime

I have no idea what you think you're saying, and, after a number such exchanges with you, my bet is that you have no idea what you think you're saying.

Edit: On second thought, I do see your point: The uncountable co-idemponality topos decountably isomporphizes with the muti-hyperprimes to induce a Booleantype semi-constructive syntax field.

Yes, how could I have missed that?

I think saying "there exists a set of all natural numbers" is equivalent to writing a program to print all natural numbers and running it through to completion. — keystone

You've described your notion of potential infinity a few times (in another thread especially). And I've replied about it each time. Now, you're coming back to restate it, but still not addressing the substance of my previous replies. As in another thread, this just brings us around full circle.

However, I think set theory can be reframed to correspond to potentially infinite algorithms instead of actually infinite — keystone

Same as above.

I'm not convinced by your response to Thompson's lamp because your answer lies outside of the thought experiment — keystone

The thought experiment is suggestive of an analogy with set theory, but suggestiveness is not an argument about set theory itself. One sets up a thought experiment, then suggests an anology with set theory, as that analogy however is only informal and imaginary. Then I immediately concede that set theory doesn't explain informal, imaginary analogies to it. The terms of the thought experiment are not set theory.

Set theory doesn't have lamps that turn off and on infinitely but such that there's a final state in which the lamp is either off or on.

But set theory does have infinite sums if there is convergence. So, set theory does not say there is such a "final state" for a non-converging sequence of 0s and 1s. Set theory doesn't have a contradiction that there is a final state for such a sequence. That's something good about set theory.

But one can say, what about the fact that set theory has the finite ordinals, but then the least infinite ordinal that comes after all the finite ordinals? Yes, but no one says that there is "process" by which we go trough all the finite ordinals and then arrive at the least infinite ordinal.

why would we think they can be completed in reality? — keystone

We don't! Set theory doesn't say there's a "completion in reality". Set theory doesn't have that vocabulary.

Since one cannot disprove the existence of an infinite being (e.g. God) one cannot disprove the existence of infinite sets. — keystone

Your analogy betwen mathematics and theology is not apt.

One can disprove 'there exists an infinite set' by stating axioms that disprove 'there exists and infinite set'. The obvious choice for such an axiom is 'there does not exist an infinite set'.

Anyway, I never asked you to disprove anything at all.

infinite universes (in which infinite sets can exist) harbor contradictions — keystone

A contradiction is a certain kind of sentence. But, of course, there is no world in which a contradiction is true.*

* The context here is ordinary logic.

And there is no contradiction in set theory.*

* As far as we know.

We don't intend or claim that a domain of discourse for set theory is a world such as a physical world of physical particles and physical objects. At the beginning of this discussion, if asked, I would concede that immediately.

Now, if you wish to have a mathematical theory, adequate for science, that does have a domain of discourse of physical particles and physical objects, then no one is stopping you from saying what that mathematical theory is, or might be, or some idea of it. Saying, [paraphrase] "We'll keep set theory except infinite sets and use potential infinity instead" is suggestive of an idea, but not much more.

where infinite processes can and cannot be completed. — keystone

I know of no theorem of set theory that there are infinite processes that both can and not be completed.

The "paradox":

Let there be a hotel with denumerably many rooms with room numbers 1, 2, 3 ...

Suppose there are denumerably many guests in rooms and that each room has a guest.

Suppose a new guest arrives. Then the hotel manager moves the already staying guests this way: If a guest is in room j, then that guest is moved to room j+1. And the new guest is put in room 1.

Or suppose k (k a natural number such that k>1) number of new guests arrive. Then, for all guests, if the guest has been in room j, then the guest moves to room j+k. And the k new guests are put in rooms 1 through k.

/

Indeed it is the failure of the pigenonhole principle for infinite sets (i.e. that infinite sets are Dedekind infinite) that allows the "paradox".

A perpetually growing hotel — sime

As I recall, it's not a perpetually growing hotel. Rather, it's a hotel with denumerably many rooms and denumerably many guests, one to each room.

The "paradox" is not about potential infinity, but rather about the set of natural numbers (or any denumerable set along with a given enumeration of it).

But such a hotel isn't describable in ZF if the axiom of choice is assumed, because it forces Dedekind-infiniteness upon every infinite set. — sime

You have it backwards. That the set of rooms is Dedekind infinite is what makes the hotel "paradoxical".

Moreover, we don't need any choice axiom to prove that the set of natural numbers is Dedekind infinite.

Hilbert's Hotel refers to the trivial possibility of indefinitely expanding a finite hotel — sime

The hotel is not finite. It has infinitely many rooms.

ZFC cannot distinguish between a hotel that isn't finite purely because it is growing without bound, from a mythical hotel with a countably infinite subset — sime

ZFC doesn't distinguish among hotels, real or mythical.

And the point is not that there is a countably infinite subset. Every countably infinite set has a countably infinite subset.

the fault of the axiom of choice — sime

I don't see what the axiom of choice has to do with it. The rooms are enumerated by room numbers. Choice is not invoked.

"A Hilbert Hotel has a countably infinite subject" refers to a sentence of ZFC, and not an actual hotel. — sime

Hilbert's Hotel is an imaginary analogy to ('S' for the set of positive natural numbers):

For any natural number k>0, S is 1-1 with S\{1 2 ... k}.

For k = 1, as a new guest arrives, we move each already staying guest to the room above.

For k>1, as k number of new guests arrive all at once, we move the already staying guests up more than one room, as we move them up k number of floors.

/

For me, the problem is not so much that there is anything counter-intuitive about this, but rather that it's rude and bad business practice to keep waking guests up in the middle of the night and make them pack and move to another room, especially an infinite number of times. Not only that, but the poster keystone has added lamps that keep turning off and on, which is extremely annoying when people are trying to get a good night's rest for the next day when everybody is going out to see Zeno's 10K Charity Run where Achilles will have to run through an infinite number of distances and suffer the ignominy of getting beat by a turtle.

The T schema doesn't dictate

1) The type of truth object (sentences vs propositions)
2) The nature of the equivalence relation (analytic necessity vs material necessity vs modal necessity)
3) Whether the schema is used prescriptively to exhaustively define the meaning of "truth" e.g as in deflationary truth, or whether the schema is used to non-exhaustively describe truth but not explain the truth predicate, as in inflationary truth. — sime

1) Tarski was dealing with sentences.

2) The biconditional is for material equivalence.

2) It's a definition.

I don't have answers to those questions.

TonesInDeepFreeze will be in the know as to how far Tarski's usage (as clarified above) is agreed, in modern logic and maths related discourse. — bongo fury

I know some of the basics of mathematical logic pretty well, but I'm not even within a telescope's distance of being an expert on it or anything other than jazz, and even on that subject I'm outclassed by a number of greater experts.

It doesn't have to conform to ordinary linguistic usage in this narrower technical one. — bongo fury

I am using 'denote' in the everyday sense:

"stand as a name or symbol for"

'denotes', 'names', 'stands for', 'symbolizes'. All good.

'Chicago' denotes Chicago.

Chicago is denoted by 'Chicago'.

The denotation of 'Chicago' is Chicago.

In the most ordinary context of mathematical logic:

An interpretation (model) is a certain kind of function from the set of symbols. The function maps a symbol to an element of the domain, an n-ary total function on the set of n-tuples of the domain or an n-ary relation on the domain, depending on the kind of symbol.

Would you please tell me what textbook(s) in mathematical logic is the source of your understanding of the basics of this subject? Then, if I have the book, I can see better how we can at least agree on the basics.

I don't think this term is strictly grammatically correct — RussellA

'x' denotes y.

That is grammatically correct.

That usage is well established in everyday language and in logic.

Denotes infers points to — RussellA

That is not correct.

The word 'denotes' doesn't infer. People infer; words don't infer.

I still believe that "snow" is denoted as snow is grammatically correct. — RussellA

I'd like to see somewhere an example of

'x' is denoted as y.

If I saw someone write that, then I wouldn't know what it is supposed to mean:

(1) 'x' is denoted by y.

Which is backwards.

or

(2) y is denoted by 'x'.

Which is correct.

Your Cambridge examples are about relations between things: fetal heart rate and percentage of time. We're talking about the relation between a word and a thing.

In the world there are no paradoxes. — RussellA

Two separate things:

(1) An inconsistent theory has theorems that are contradictions. That is syntactical.

(2) There are no true contradictions. That is semantical.

To avoid paradox in language we need to ensure that language corresponds with the world, because the world is logical.
Tarski is aiming at the same goal. — RussellA

In the kind of formal theories most pertinent to this subject, contradictions are stated in a language, but the concern is whether they occur in theories. Tarski is not worried about whether languages are consistent, since 'consistent language' doesn't even make sense. What are consistent or not are theories.

I don't know where Tarski is supposed to have couched anything he said in a way similar to what you wrote. I don't know where Tarski is supposed to have said anything fairly paraphrased as "the world is logical". What are logical or not are inferences and arguments.

the definition does not lead to paradoxes and it is not circular. — RussellA

Right. And the way that is achieved is by adhering to certain definitional forms with certain restrictions.

IE, paradoxes in language may be avoided by ensuring that language corresponds with a world that is logical. — RussellA

No, you are adding that, from your own notions. Unless you show me exactly what Tarski wrote that can be reasonably paraphrased as "ensuring that the language corresponds with a world that is logical". Again, where does Tarski say "the world is logical"? What would he mean by it when what is logical or not is not a world but rather an inference or argument? And where does Tarski say a language corresponds with the world. What corresponds or not with a world is a sentence or theory.

It is true that Tarski does not say that white is a necessary condition for snow. — RussellA

Does Tarski even say that snow is white? In the "standard interpretation" of 'white' and 'snow', we get that 'snow is white' is true. But it is not precluded that the words can be interpreted otherwise. I went into detail about that in my previous post.

Tarski uses the analytic proposition "snow is white" — RussellA

Please say where Tarski says 'snow is white' is analytic.

I really do not want to get bogged down into an endless debate about analytic/synthetic, but you are taking a leap with it here regarding Tarski, unless you show us where he said that 'snow is white' is analytic.

rather than a synthetic proposition such as "snow is always welcome" . — RussellA

He could have used that also.

We say for any statement P,

'P' is true iff P.

Whether P is synthetic, analytic, or supercalifragilistic:

'P' is true iff P.

'Bob's car is white' is true iff Bob's car is white.

That's another instance of the schema.

The schema does not at all require that its instances be analytic sentences.

You wrote - "snow" is precipitation ..............white...............
You didn't write "snow" is precipitation.........which may or may not be white........ — RussellA

That's a good point. I overlooked that I chose a definition that happened to include 'white' in the definiens of 'snow'. That was a mistake. I don't know whether Tarski even had a scientific definition of 'snow' in mind, and especially one that has 'white' in the definiens. So, I don't know whether Tarski thought of a particular definition so that he regarded 'snow is white' as analytic. I highly doubt that he did.

I should have chosen one such as this:

"precipitation in the form of ice crystals, mainly of intricately branched, hexagonal form and often agglomerated into snowflakes, formed directly from the freezing of the water vapor in the air"

The point of the Tarski schema is not to define 'is true' for just analytic sentences.

"grass: vegetation consisting of typically short plants with long, narrow leaves, growing wild or cultivated on lawns and pasture, and as a fodder crop"

So 'grass is green' is not analytic.

'grass is green' iff grass is green.

"snow" is frosty stuff and "white" is the colour of St Patrick's Day T-shirt are external — RussellA

The interpretation of the words takes place in the metalanguage.

What is your definition of 'external'?

From the IEP - The Semantic Theory of Truth
"A standard objection against STT points out that it stratified the concept of truth. It is because we have the entire hierarchy of languages — RussellA

Correct. To define 'is true' for sentences in a metatheory, requires a meta-metatheory, ad infinitum.

But that doesn't entail that for the object language we can't both interpret it and state the definition of 'is true' in the metatheory. In ordinary mathematical logic we do give interpretations of an object language and also state the definition of 'is true' per each interpretation - all in one metatheory. (Or is there something I'm overlooking?)

Yes, within a particular MML, there is only one interpretation. Between different MML's there are different interpretations. — RussellA

Let L be the object language.

Let x(y) be the metalanguage for y.

m(L) interprets L

mm(L) interprets m(L)

ad infinitum

There are not clashes of interpretation.

The axiom 1 + 1 = 2 exists within a Metametalanguage (MML) — RussellA

An axiom is written in a language but occurs in a theory.

For example, the axiom

Ax(x+0 = x)

is written in the language of first order PA and it occurs in first order PA.

The language of a theory and the theory itself are different things.

Is it a coincidence that the word "irrational" means illogical/makes zero sense? I recall starting a thread on how irrational numbers could be the smoking gun that there's something seriously wrong with mathematics and the universe itself. — Agent Smith

You have got to be kidding! Please tell me you are kidding!

Gaudeo te relinquere, domine.

There, see, I took you up on your suggestion to visit Google.

It's not so much that you don't meet a standard of rigor, it's that you lie about the subject.
— TonesInDeepFreeze

:rofl: — Agent Smith

You'd be right if that emoticon meant 'QED'.

Ok! Google should help you find the relevant pages! Bonam fortunam. — Agent Smith

What? You're the one spreading disinformation, notwithstanding the little touch you give with Latin phrases.

You say that set theory claims that a (proper) part can be equal to the whole. I explained to you twice exactly the way you are incorrect. My going to Google or not doesn't affect that you're spreading disinformation.

it doesn't meet the standards of rigor required in mathematics. — Agent Smith

It's not so much that you don't meet a standard of rigor, it's that you lie about the subject.

There's nothing in the world that doesn't have detractors. So, by your logic, everything is wrong.*

Finitism has detractors, so by your logic, there's something very wrong with finitism.

* Or, to put it not so sweepingly, let's unpack what you've said.

(1) As I've pointed out to you probably half a dozen times already, finitism has many forms. Some finitists work in infinitistic set theory.

(2) Many finitists are critics of infinitistic set theory. But your own criticisms are ignorant, self-malinformed, and disinformation. You don't like the concept of infinite sets and want to see a mathematics without them. That is fine. But you grab any chance to indulge your confirmation bias about the subject.

In this case, you don't even appeal to specifics but instead ludicrously reason that since there are critics then we can pretty much bet that they're right. That is so remarkably irrational. It's in the same league of irrationality as people who say, "The conclusion that climate change is anthropogenic is wrong, which I know because there are scientists who say it's wrong."

No, to claim that the conclusion is wrong requires actually comparing the work of the dissenting scientists with the work of the preponderance of other scientists.

To claim that set theory is wrong requires comparing the arguments of the detractors of set theory with the arguments of the mathematicians and philosophers in favor of and in defense of set theory.

Just grabbing random out of context quotes against set theory, one-liner tidbits, and polemical irrelevancies to argue against set theory is like viewers of Fox News who base their claims about politics on whatever chyrons happen to cross the screen, whatever infantile propaganda memes are splashed and whatever disassociated falseoids happen to spill from the mouths of the on-screen anti-pundits.

disinformation
— TonesInDeepFreeze — Agent Smith

Then you write a post having nothing to do with your quote of me. It still stands that it is disinformation to say that set theory claims that a (proper) subset of the whole is equal to the whole.

the existence of finitism suggests to me that there's trouble in (Cantor's) paradise! — Agent Smith

There are critics of X, therefore there is something very wrong with X.

That is a risibly stupid argument.

Okie dokie cranky spanky.

I take "approaches" to be a potentially infinite process. — keystone

That just takes the conversation back to where we were before. One can have whatever concept of limits one wants to have, including conceiving in terms of potential infinity. Indeed, there are systems that do (and I little doubt succeed) in axiomatizing large amount of analysis without infinite sets.

And lots of people aren't concerned with axiomatization, so they don't even care whether or how analysis is axiomatized, as they at the same time prefer to conceive in terms of potential infinity rather than there being infinite sets. Not at issue.

But, to the extent that one is interested in axoimatization, one would want to go the next step, which is to ask, okay, what are the axioms, or at least, what initial ideas are there for what the axioms might be?

If you say, that's not your concern, then fine. Then we just have different roads we want to travel. But one can't fairly criticize the road of set theory if one is not addressing it as an axiomatization. And even if not criticizing set theory but instead just saying mathematics can be done with unformalized "potential infinity" instead, then it's not a fair comparison since one is an axiomatization and the other is not.

That means you understand why it's not such a good idea to post disinformation that set theory claims that a (proper) part is equal to the whole?

And maybe, you'll say who you thinks says that set theory does make that claim?

I can think of x^2+y^2=1 without having to think of any points — keystone

Of course. No one says that you have to.

The points come from the circle, not the other way around. — keystone

A circle is a certain kind of set of points. I don't know what you mean.

No, there is no such thing as a sphere with infinite radius...that's explains why I can't imagine it. — keystone

Yes, and since infinitistic mathematics doesn't have spheres with infinite radi, infinitistic mathematics doesn't call on you to imagine it.

a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value — keystone

A sequence is a set. And it has a domain, which is a set, and a range, which is a set. An infinite sequence is an infinite set with an infinite domain.

Of course, one can leave that unconsidered, not in mind, when working in certain parts of calculus. That is not at issue. But when we trace the proofs of the theorems of analysis back to axioms, then, in ordinary treatments, those are the axioms of set theory.

the informal definitions that us engineers were taught didn't use sets. — keystone

Mathematics, in many branches, is brimming with sets. Analysis, topology, abstract algebra, probability, game theory... Can't even talk about them, can't get past page 10 in a textbook, without sets.

But of course, one can use the theorems of mathematics for engineering without tracing the proof of those theorems back to axioms, in particular the set theoretic axioms. That's not at issue.

an infinite set and its proper subset have the same cardinality — Agent Smith

Just to be clear, an infinite set has the same cardinality as some of its proper subsets, but not all of them.

in colloquial terms means a part is equal to the whole — Agent Smith

No it does not. It means a proper part is in 1-1 correspondence with the whole, not equal to the whole. A proper subset S of T is never equal to T. I've mentioned this before, but you repeat your disinformation.

there are peeps who say this is exactly what the mathematics says — Agent Smith

Who specifically?

∞ maybe the internal combustion engine of math - creates more problems than solutions. — Agent Smith

I can only guess at what vague thing you mean every time you write the leminscate.

I don't know what problem, especially a practical one, you think is caused by set theory.

Must an ordered field necessarily be a field of numbers? — keystone

No. But all complete ordered fields are isomorphic with one another. So all complete ordered fields are isomorphic with the system of reals.

Could it instead be a field of equations? — keystone

I know there are field equations (but I know really nothing about them), but I don't know what a field of equations is.

the axioms of set theory are not in concordance with the intuitive notions of 'finite sets' — keystone

All the axioms are in that concordance, except one.

And since the only sets we ever work with directly are finite, I think we should be cautious accepting axioms that oppose them. — keystone

I think there's something to that. Indeed, it is, roughly speaking, right in line with Hilbertian finitism. Hilbert's idea was that we can work in infinitistic mathematics if we have a finitistic proof of the consistency of infinitistic mathematics. Famously, we found out that there is no finitistic proof of the kinds of systems we'd like to use, not only not of set theory but even of PRA, the system itself that we may take as exemplifying finitistic reasoning at its "safest". Yet, if I understand correctly, Hilbert's condition was a sufficient condition not a necessary one.

Instead of saying 'there exists an infinite set' I would be comfortable saying 'there exists an algorithm that describes an infinite set'. — keystone

That's interesting. But, if that is to be a statement in the system, we'd need to see "described" couched mathematically. I have a hunch that your notion is pretty much the same as 'there exist potentially infinite sets', and as I've said, I don't know a system that says it.

the point — Banno

I don't take it that there is "the" point, but rather many forms of application, engagement and appreciation. Some of them not necessarily that of "putting to use" except in the broad sense that virtually any attention, even if purely aesthetic, is a form of "putting to use". Each person may find the appeal of mathematical logic on their own terms, which of course includes using it in the sense you mention, but also may be primarily enjoyment of seeing concepts of reasoning so ingeniously, rigorously and elegantly articulated even irrespective of the use in philosophy and the sciences. Then the usefulness in philosophy and the sciences is a huge added bonus.

About a meta-metalanguage:

What is wrong with this?:


Given a language L, and an interpretation M of L, and a sentence P of L:

A sentence 'P' is true per M iff P.


That's just like any textbook in mathematical logic.

No meta-metalanguage.