I had in mind transitive models. — Nagase

That works. Thanks.

I don't see now this could be a set in TF. — fishfry

TF proves there is no such set. But meanwhile set theory proves there is that set. The set is the universe for a model of TF. The set itself is not a member of that universe.

(ZF-C)+~C is just ZF+~C.

That is not ZF.

I’ve never seen that theory discussed (though maybe it comes up somewhere.)

I take ‘-‘ to mean ‘without’ and ‘+’ to mean ‘also with’.

So ZF-Inf is ZF but without the axiom of infinity.

And (ZF-Inf)+~Inf is ZF without the axiom infinity and also with the negation of the axiom of infinity.

Two very different theories.

If you mean the part where it lists the axioms only referring to the successor function S and the symbol e — fishfry

No the part that has the headline:

First-order theory of arithmetic

That is first order PA. (In this context, By ‘PA’ we mean first order PA.)

And PA does not have set abstraction notation.

Here is what I responded to:

The universes of ZF-Inf are all infinite. This is clear from the fact that ZF-Inf has the power set axiom, so that there's no bound for the size of its sets. — Nagase

When you say "its sets", I take it you mean the universes's sets, i.e. the sets that are in the universe. Of course, for any particular universe, there is a bound on the sizes of the sets. So I took it that you meant that for any set size there are universes for the theory that have members of that set size. And I agree. My point is that this holds for any theory whatsover, regardless of its axioms or, in particular the power set axiom.

if a given model has cardinality n, then it has no members of cardinality n+1 — Nagase

If by 'members' you mean members of the universe of the model, then the above is incorrect. Trvially, let the universe of some model be {2}. The universe has cardinality n=1 but it has a member of cardinality n+1. And beyond the trivial, for infinite universes, it is not precluded that a universe is of cardinality C but the universe has a member of cardinality C+1, where C+1 is the successor cardinal of C.

I'll use M, because it stands out better.

So:

For a theory T with 'e' in the language, a formula F(x) in the language for T "invokes" a proper class relative to T
iff
(i) T proves Ex F(x). (ii) T proves ~Ey(Mx & Ax(F(x) -> xey)).

But that depends on the theory having M as primitive or defined (or being in a 2-sorted logic) And M not to be just an arbitrary predicate, I guess the theory has to prove Ax(Mx <-> Ey xey).

So:

For a theory T with 'e' in the language, and having a 1-place predicate symbol 'M' as mentioned below, a formula F(x) in the language for T "invokes" a proper class relative to T
iff
(i) T proves Ex F(x). (ii) T proves ~Ey(Mx & Ax(F(x) -> xey)). (iii) T proves Ax(Mx <-> Ey xey).

This is not a problem for any language with 'e', as such an 'M' can be defined if not primitive.

So 'x is an ordinal' invokes a proper class relative to NBG.

But PA is not eligible for applying this definition, because PA does not have 'e'. So what about (ZF-Inf)+~Inf as a surrogate for PA?

We want to see whether 'x is a natural number' invokes a proper class relative to (ZF-Inf)+~Inf.

I guess it works. (?)

I mentioned (putting it in these terms now) that for any theory T and any cardinality C, if there is a model M of T, then there is a model M* of T such that the universe for M* has a member of cardinality C. (Though originally I forgot to mention the obvious qualification that this pertains when there is a model M of T.)

You asked me what I had in mind. So in my previous post, I gave the proof.

By the way, since historically different writers formulate class theory differently, for sake of definiteness, I choose one in particular: ‘Model Theory’ by Chang and Keisler.

We should not overlook that ‘class’ does not mean just ‘proper class’. Some classes are sets and other classes are proper classes. Everything in NBG is a class.

Yes, NBG does not prove that there is a set of all the ordinals. To be clear, with NBG we don’t have variables for just proper classes. With the multi-sorted version we have general variables for all objects - all classes, i.e sets and proper classes. And we have another special variable for just sets. Where ‘x’ and ‘y’ are general variables (class variables) we have the theorem

EyAx(x is an ordinal -> x e y)

And where ‘s’ is a set variable, we have the theorem

~EsAx(x is an ordinal -> x e s)

As you mentioned, having two kinds of variables is equivalent to having a primitive predicate M for ‘is a set’ and relativizing. (But a primitive predicate C for ‘is a class’ turns out to be the universal predicate: Cx <-> x=x.) So the second theorem is

~Ey(My & Ax(x is an ordinal -> x e y))

But the first theorem itself shows that your formulation doesn’t work unless you revise it in some way, perhaps with such relativization, and we would have to see exactly what that formulation would be.

notation: for a function f, let f’y= the x such that fx =y.

Let M be a model for the language L.

Let S be any set whatsoever with the same cardinality as the universe U for M. Doesn’t matter the cardinalities of the members of S. Let f be a bijection from U onto S.

Then, for example, suppose ‘H’ is a relation symbol in L, and suppose M maps ‘H’ to the relation R in UxU.

Let M* be the model with universe S, and, for example, M* maps the symbol ‘H’ to {<j k> | <f’j f’k> in R} ... and in that way for the rest of the signature of L.

So M and M* are isomorphic.

Or did I make a mistake?

You more or less despise "my world", not knowing anything about it other than it includes people who may recommended books without first knowing whether they are also available online.

Because I recommended some printed books (which you may check for yourself whether they are also online or not), you infer that I don't know enough about HTML and should read a book about it? That would prevent me from ever again grievously recommending books when I don't know whether they are also available online?

you and your friends — alcontali

What friends? You said I don't have any.

Whatever may be the demerits of printed books, at least I an tell you that those are exceptionally great books and it is not certain that there will be better ones online.

But did you even look to see whether these might also be online?

what Gitman wrote, is an extension of the signature, and therefore of the language of the theory — alcontali

Exactly.

paucity of materials online — alcontali

I don't know about online, but here is a course of books I highly recommend, in order of study:

* Logic: Techniques of Formal Reasoning - Kalish, Montague, Mar

This tells you how to make sure that your mathematical arguments are within the first order predicate calculus. But if you feel that your mathematical reasoning is not prone to such things as mistakes with quantifiers and improper instantiations of variables, then you can skip this book. Though I still highly recommend it to make really sure you're always on firm ground.

* Elements Of Set Theory - Enderton

From the previous book, you will have a good grasp of proof in the first order predicate calculus you need for a rigorous study of set theory.

* A Mathematical Introduction To Logic - Enderton

From the previous book, you will know the set theory you need for a rigorous study of mathematical logic.

These infinite cardinals — alcontali

Yes, as I said, if a theory has an infinite model, then it has both countable and uncountable models.

Therefore, Gitman's "language" does not always have a countable number of sentences — alcontali

That's a non sequitur. The set of sentences is countable but, if there is an infinite model of the theory, then there are countable and uncountable models of the theory.

And Gitman goes on to discuss particular countable models.

The language, the theory, and the model are separate, but related things.

If you like, I can recommend introductory textbooks in this subject that will explain all this for you, step-by-step, in greater detail and context than I can give in posts.

Expand LA by adding a constant c to obtain the language LA∗ = (+,·,<,0,1,c). — alcontali

She's adding a constant to the LANGUAGE of the theory. The model is a model for the language. The model is not a language.

There is a language. Then there are two things.

(1) A model for the language.

(2) A theory in the language.

Then the model is a model of the theory iff (the theory is in the language, and the model is a model for that language, and every sentence in the theory is true in the model).

a theory, i.e. a set of axioms — alcontali

Writers might differ on the definition of 'theory', but it turns roughly the same with whatever adjustments we need:.

(1) A theory is a set of sentences closed under deduction. The theorems of the theory are the members of the theory.

That definition has the advantage that we can consider theories without having a particular set of axioms in mind. For example, given a model M, we have Th(M), which is the set of sentences true in M. So an axiomatization doesn't have to be specified.

(2) A theory is the set of theorems derivable from a particular set of axioms.

With that definition, given a set of axioms S, the theory axiomatized by S is the set of sentences derivable from S.

a model, i.e. a CS-like grammar describing sentences — alcontali

I don't claim that is not a notion in computer science, but in context such as Gitman, which is set_theoretic/mathematical_logic/model_theory, that is not how the concept of a model is formulated.

countable models, but I have to leave it open if the concept properly applies to models that are larger than that, such the nonstandard models of PA. — alcontali

The particular nonstandard models of PA that Gitman is discussing are countable models. There are countable nonstandard models of PA and uncountable nonstandard models of PA. And, in the first definition of 'model of' I mentioned in the earlier post, the language is countable, and with the second definition, an uncountable language for a model of PA can be restricted to a countable language.

it would be useful for Gitman to explain why such uncountable nonstandard model would still be a legitimate language. — alcontali

A model is not a language.

An uncountable model is one in which the universe for the model is an uncountable set. Every theory that has an infinite model has both countable and uncountable models, even as the language for the theory is countable.

an actual non-standard model of PA is a model for the language of PA, not a model for a different language with an additional symbol. — GrandMinnow

What I wrote there (and with similar remarks on this particular point) may be too strict. It belies that there may be a less narrow notion of 'model of'.

Narrow:

A model M of a theory T is model for the language of T, such that every sentence in T is true in M (a theory being a set of sentences closed under deduction). So a model of PA is model for the language of PA (which does not include an added constant 'c'). So a model M that, for example, Gitman proves to exist is not itself for the language of PA and therefore not a model of PA, but then we restrict such a model to the language of PA and that restriction is a model of PA.

Less Narrow:

We could allow that a model M of a theory T is model for a language that includes the language of T, such that every sentence in T is true in M. So a model M that, for example, Gitman first proves to exist is itself a model of PA, and we don't need to then restrict the the model.

The term "language" in her slides is something that can have an uncountable number of sentences. — alcontali

She doesn't preclude uncountable languages in general. But where she proves the existence of non-standard models of PA, the languages mentioned are countable languages. Uncountability does not have a role in it.

That’s not first order PA. Look further down in the article where the axioms of first order PA are listed.

"Read off"? Is that a technical term? — fishfry

No. I was giving practical advice to not overlook that when we read natural language renderings of formulas, then we can't expect that how we naturally take such locutions in English is preserved with every interpretation (model) for the formal language.

suppose you adhere to something like the limitation of size conception, according to which sets are collections that are not too big (say, are not the size of the universe) — Nagase

I don't begrudge anyone from that notion, but, for me, it's too vague. What is "too big"? And which universe?

Perhaps we can say that a collection is a proper class relative to some theory T if: (i) there is a predicate P such that x belongs to the class iff P(x), (ii) T proves that there is an x such that P(x) but (iii) T proves that there is no y such that x belongs to y iff P(x). — Nagase

I might suggest saying (only a subtle difference with yours):

For a theory T with 'e' in the language, we say a formula F(x) in the language for T "invokes" a proper class relative to T
iff
(i) T proves Ex F(x). (ii) T proves ~EyAx(F(x) -> xey).

So there is no proper class mentioned there. Instead there is a two place relation between a theory T and a formula F. Okay, fair enough, as maybe that's what fishfry has in mind. And it does work where T is set theory itself, indeed as it is common even in informally stating such things as the axiom schema of replacement where we refer to a 'function class'.

But I warn against thinking conflating this with the one place predicate 'is a proper class', except quite informally.

But worse, this manifestly clashes with ordinary terminology.

Let T = NBG (I call it 'BC' for Bernays class theory). For example:

BC proves Ex x is an ordinal. But BC does not prove ~EyAx(x is an ordinal -> xey), indeed BC proves the negation of that. But we do say, with BC, that the class of ordinals is a proper class.

Moreover, second-order PA can't be at issue, because second-order PA proves that there is an infinite set, namely the set of all natural numbers, so the natural numbers cannot be a proper class relative to this theory. — Nagase

Yep.

PA doesn't have sets, and second, even if it did that would not be a valid specification of a set since it violates the axiom schema of specification. — fishfry

Hard to discuss a counterfactual here.

So let's turn to TF.

It's not a matter of being consistent with the axiom schema of specification.

Instead, in the absence of the axiom of infinity, we do not have a supporting existence theorem for a definition:

N = {x | x is a natural number}

Read further down in the Wikipedia article, and you will see the axioms for first order PA. There is no predicate 'is a number'.

/

Regarding your notion about improper sets relative to PA as personal visualization, I didn't ignore it - you even quoted me remarking on it. I said I don't opine as to what does or does not make sense in your mind. But I said your notion makes no sense to me. And I would add that I think it does muddle discussion. But I didn't say you shouldn't think it.

ZF-infinity means ZF plus the negation of the axiom of infinity by default. — fishfry

Some people use it that way. And I have seen it lead to misunderstandings as casual readers fail to note that, in this context, we need the negation of the axiom of infinity and not just leaving out the axiom of infinity.

it's all the finite sets — fishfry

No, not all the finite sets. Only the hereditarily finite sets.

You just explained to me that HF are all the finite sets in ZF — fishfry

I defined 'HF' to stand for the theory (ZF~Inf)+~Inf.

But now I realize that writers often use 'HF' to stand for a class. So my choice to use 'HF' as the abbreviation was not good. From now on, I won't use it to stand for the theory (ZF~Inf)+~Inf. Instead I'll use:

TF = (ZF~Inf)+~Inf

since N is not a definable symbol in PA, I can't say "N is a proper class" because I have no idea what N is. Is that right? — fishfry

To be more precise, whatever symbol 's' we pick, TF does not support a definition:

s = {x | x is a natural number}

because the theory does not prove that there is a such an object.

your definition [of 'proper class' makes perfect sense. I do wonder why I haven't seen it. — fishfry

In class theory, it is well understood that a proper class is a class that is not a member of any class. All I'm doing is pointing out that we can also say that in set theory and conclude in set theory that there are no proper classes. It might be annoying, because it's not a very useful series of formulations. But it its technically correct, and I find that it sharpens the picture. Especially it goes against a common misconception that we can define a predicate symbol only to stand for a relation (sets are 1-place relations) that has members. No, we can always define an empty predicate. For example:

dfn: Jx <-> (x is odd and x is even)

is allowable, even if rather pointless.

That's not a model of PA. w+2 has no successor. — fishfry

Was a typo of omission; I meant {w, w+1, w+2 ...}

"is a natural number" must be a predicate [...] Therefore I can form the COLLECTION, or "predicate satisfier," or as it's officially called the extension of the predicate, N = {x | x is a natural number}. N is a class and it's not a set. So it's a proper class. — fishfry

Yes, we can have a predicate 'is a natural number' in TF. And upon an interpretation of the language, it has an extension (a subset of the universe for the model) and that extension is a set, not a proper class.

Moreover, indeed Gitman, just as the Wikipedia article, uses the method of adding a constant as a step in the proof of existence, but then she says explicitly that a non-standard model of PA is a model of the axioms of PA.

The axioms of PA do not have the added constant. There is a difference between (1) a proof of existence through a detour via another language and (2) an actual model of PA.

I didn’t say there is anything wrong with Gitman. You seemed to have skipped what I said about it. Actually, I should be sharper by saying that it appears to be a step in proving existence - and, again, an actual non-standard model of PA is a model for the language of PA, not a model for a different language with an additional symbol.

And what you wrote is not, as you are claiming, literally what she wrote. She mentions an arbitrary constant ‘c’. But you introduced in particular ‘w’ (omega), so I would think you have in mind omega (the set of natural numbers).

So, by w-1, do you mean some kind of ordinal subtraction, or something else?

It’s not vitriolic to point out that you are posting misinformation and that you would do better to read a textbook.

That you posted a topic doesn’t provide that people should not correct your misinformation nor advise that you wouldn’t be so prone to confusions if you read a textbook.

I should add that study of non-standard models of PA usually considers not merely that the models are not isomorphic to the standard model but also the “blocks” of linear orderings. And, yes, these do resemble the standard ordering of the integers (negative and non-negative). But I took your w-1 to mean omega minus one - maybe yoh didt mean that?

And I looked at the Wikipedia page you referenced. The method of adding a constant is a step in proving the existence of non-standard models of PA. But the actual non-standard model is on a language that does not include the constant.

A non-standard model of PA I guess can be visualized through the method Gitman mentions, but literally a model of PA is itself a model for the language of PA, not for a language with an added symbol. The elements with 'c' in her slides are just elements of the universe. Again: A non-standard model of a theory is a model for the language of the theory (not with added symbols or sentences) such that the model differs from (or is at least not isomorphic with) some agreed upon standard model.

That you reference those slides to misunderstand them, is a function of relying on various Wikipedia articles and Google searches for various presentation, out of context of a systematic treatment such as in a good textbook.

This is typically represented by adding a symbol w — alcontali

No, you have it very wrong.

The universe has additional members. We don't add symbols that engender sentences not already in the theory.

You are completely confused about this and other topics in mathematical logic. Please stop and instead first read an introductory textbook in the subject.

if you do not want to make constructive remarks — alcontali

My remarks have corrected your misinformation. That is constructive. And a further constructive remark is that you need to read a basic textbook in the subject instead of spreading wanton misinformation about it.

Anyway my question stands: In the earlier passage, what are you talking about? Model theory, computer science or your own blend of them?

And I have no idea what you mean by w-1 and w-2 in the context of saying w-1, w-2, w, w+1, w+2. Do you have a definition of some kind of operation of ordinal subtraction that allows subtracting 1 or 2 from a limit ordinal? What exact set or ordinal type do you mean by w-1 and w-2? Actually, nevermind, better you should instead get a book on working in the first order precicate calculus, then an intro book on set theory and then an intro book on mathematical logic.

They are useful fictions. — Relativist

I don't opine whether they are fictions or not. But, for me, at least such views as fictionalism and instrumentalism that allow sets as consistent fictions is enough for working in mathematics.

I don't opine on that particular philosophical position. But the outcome of it doesn't preclude existential quantifcation in mathematics or working with infinitistic set theory.

the question is: is there something that exists in the real world that maps to an infinite set? — Relativist

That raises the question, "What do you mean by 'the real world'"? And what do you mean by "something exists in the real world"?

Anyway, whatever the answer about infinite sets, mathematical statements themeselves are things like "There exists the set of natural numbers" or "There exists an x such that, for all y, if y is a natural number then y is an element of x" or "With these axioms we prove the formula 'There exists the set of natural numbers'" and not "There exists the set of natural numbers in the real world (whatever "in the real world" might mean)."

You want mathematics not to claim that there exist infinite sets; you want to "reject existence", as I understood you, such as existence asserted with the existential quantifer. But you haven't answered my point that without infinitistic set theory, axiomatizing the mathematics for the sciences gets a lot more complicated.

The presence of such questions doesn't impugn existential quantification.

reject its existence — Relativist

Of course, we can hold that there do not exist infinite sets. But then providing a formal axiomatization for the mathematics for the sciences gets a lot more complicated.

In mathematics and logic, we use the existential quantifier(∃) in a way inconsistent with the above — Relativist

Existential quantification is not inconsistent with the claim that abstractions are not material objects.

Now we get to infinity. The arguments I've seen for its existence — Relativist

In set theory (since you mentioned Cantor, who provided the main pre-formal concepts) there is not an entity called 'infinity' (distinct from a other notions such as points on a real extended line or figures of speech such as "as x goes to infinity"). Rather, there is the adjective 'is infinite', and an axiom that entails (with other axioms) certain theorems including the existence of infinite sets.

Infinite-size models cannot be handled by any grammar, because their sentences are necessarily countable, while these models may not have a countable cardinality. It can only work for something like ZF-inf, of countable cardinality. — alcontali

That's computer science, or a special bland of computer science and model theory that you are working out live?

throwing that symbol into the fray, won't make it work — alcontali

Adding a symbol is not relevant, not due to whatever you said about sizes of models, but rather more simply that it is not even involved in the notion of a non-standard model.

The confusion is merely in the lack of formality — alcontali

Very much not merely a lack of formality. It is dreadful confusion and misinformation.

what you write, does not necessarily help anybody to understand the subject either. — alcontali

I try to write correct statements, as best I can within the limits of posts. Much of that is merely point blank stating what is correct without necessarily giving an explanation, let alone an entire explanation. I cannot, in the space of posts, wind backwards through the trail of concepts to primitives for every concept. But at least, most importantly, I try not to write misinformation.

For actual understanding, one needs to read the basic textbooks in the subject, not just posts in a forum.