If, a big if, there did exist a finite number Nmax that could stand in for, salva veritate,∞, we could prove/disprove all mathematical conjectures via proof by exhaustion — Agent Smith

It might be the case that an ultrafinitist system would not be subject to the incompleteness theorem? I don't know. We'd have to see the system or at least a sufficient statement of its relevant properties.

/

Nothing can be understood to stand in for what you think the leminscate stands for, unless you say what you think the leminscate stands for.

Is the best course just to drop the term, then, as the nominalist recommends? Was my point originally. — bongo fury

With no comment on nominalism, I think you're right that it is cleaner not to drag in 'property'. But my original use of 'property' was not meant in a philosophically technical sense, but in an everyday sense to emphasize how the right side of the biconditional is substantively different from the left side (i.e. to highlight that the definition is not "circular"), and to do that without invoking set theoretic notions that are not everyday notions. But your suggestion is better for rigor and crispness.

Church says, "A property, as ordinarily understood, differs from a class only or chiefly in that two properties may be different though the classes determined by them may be the same (where the class determined by a property is the class whose members are the things that have that property). Therefore we identify a property with a class concept, or concept of a class in the sense [mentioned earlier]. And two properties are said to coincide in extension if they determine the same class." [italics original]

I take that to be philosophical explication.

Typo? — bongo fury

Yes. I fixed it now. Thanks.

You say.

functions (argument forms) such that if the inputs are truths (the premises), the output is a ... further ... truth (the conclusion) — Agent Smith

I might see what you're getting at, but you've not put it in a way that is clear to me.

We should take it in steps:

(1) Df. An argument is an ordered pair <G C> where G is a set of sentences (the set of premises) and C is a sentence (the conclusion).

(2) Df. A set of sentences G entails a sentence C iff for every model M, if all the sentences in G are true in M, then C is true in M.

(3) Df. An argument <G C> is valid iff G entails C.

(4) The pumpkin market has been seasonably slow. Farmers have been turning to other crops.

How does a system of logic handle truth/falsity? — Agent Smith

The standard method is the method of models

Consistency: The law of noncontradiction (LNC). A truth may not entail a contradiction (p & ~p) for if ut does, it can't be a truth. — Agent Smith

The notion of 'consistency' is purely syntactical, it does not mention 'truth'. You've added past the definition (I don't know why you do that; why you wouldn't just take a standard definition as it is written without imposing other stuff on it). The definition is:

A set of sentences S is consistent <-> S does not prove a contradiction

An equivalent definition (since first order logic has explosion):

A set of sentences S is consistent <-> there is sentence not derivable from S

So the definition of 'consistent' is syntactic not semantic.

The one sense in which you're close to something (though not part of the definition of 'consistency') is that if a sentence is true in any model then the sentence does not entail a contradiction. The reason is that entailment, by definition, preserves truth, and a contradiction is false in every model.

Some compound statements are tautologies, true always, not semantically, but solely due to logical form e.g. (p v ~p) [the law of the excluded middle — Agent Smith

The more general notion is of sentences being validities:

A sentence P is a valid <-> P is true in every model.

So the definition of 'valid' is semantic, not syntactic.

(Note: 'the sentence is valid' can also be said as 'the sentence is a validity')

Some authors use 'tautology' for 'validity', but other authors (most?) say the tautologies are the validities that are valid by virtue of evaluation of their connectives alone. (And since sentential logic is decidable, it can be semantic evaluation or syntactical evaluation. In other words, we can look at the syntactic structure or we can look at the truth tables.)

But (the context here is first order logic), due to the soundness and completeness theorems, a sentence is a valid iff it is provable from logical axioms alone. So the set of validities is the set of theorems of the pure first order predicate calculus.

However, for predicate logic that is at least dyadic, there is no algorithm to test for validity, which is to say there is no algorithm for checking the form of sentences to see whether they are valid. However, in sentential logic and in monadic predicate logic there are such algorithms.

(1) 'finitism' has different senses.

(2) Perhaps it is not necessary to have infinite sets for an axiomatization of mathematics for the sciences. It's just that in order to evaluate a non-infinitistc axiomatization, we need to have it specified.

The T-sentence is intuitively appealing as a definition of truth for natural languages — Banno

If I'm not mistaken, the 'snow is white' example was mainy to illustrate, while the main application of the Tarskian schema is for formal theories.

Yes, with that algorithm, at no step is there generated an enumeration of all the natural numbers.

But that doesn't prove that there does not exist a set whose members are all and only the natural numbers or that there does not exist an infinite set.

And it doesn't prove even the weaker claim that it is not provable that there exists a set whose members are all and only the natural numbers or that it is not provable that there exists an infinite set.

The non sequitur is:

R does not provide us with W, therefore there is no T that provides us with W.

One might as well say, "A hammer won't lift a beam, therefore nothing will lift a beam".

If we want for it to be provable that there does not exist an infinite set, then we need axioms to do that.

If we delete the axiom of infinity from ZFC and add the negation of the axiom infinity, then we prove that there does not exist an infinite set. But that theory is inter-interpretable with first order PA, which does not provide a calculus for the sciences.

And if we merely want for it not to be provable that there does not exist an infinite set, then we merely need to delete the axiom of infinity from ZFC, and then we can't prove that there exists an infinite set and we can't prove that there does not exist an infinite set. And that theory does not provide a calculus for the sciences.

This is to say that it is fine to mention the well known ostensive illustration of the notion of 'potential infinity', but that's all that is - an ostensive illustration of a notion; it doesn't even hint at how we would make a theory from it.

How is that substantively different from Thompson's lamp?

I already responded regarding Thompson's lamp.

/

I don't know a theorem of set theory that is rendered as "infinite processes can be completed".

Set theory doesn't axiomatize thought experiments.

I think it's quite easy to imagine a closed finite universe, for example a sphere of finite radius. It's a lot harder to fit in one's mind a sphere of infinite radius. — keystone

A sphere has infinitely many points in it.

And is there such a thing as a sphere with an infinite radius? If I'm not mistaken the radius of a sphere is a real number, right?

by and large I am completely convinced that the vast majority of modern math would retain it's value even IF actual infinities were banished. — keystone

We've come around full circle. Now circling for another orbit:

To axiomatize mathematics sufficient for the sciences, without infinite sets requires not just deleting the axiom of infinity from (Z\R)+CC (ie. Z without Regularity but with Countable Choice), but a very different system. There are systems without infinite sets, but I don't know how well they do or how easy they are to understand and use.

as actual infinities (rather infinitesimals) were banished from (mainstream) calculus and replaced with potential infinities (through limits) — keystone

No, limits use infinite sets. The standard axiomatization of analysis is ZFC. Ordinary modern analysis is decidedly infinitisitic. Maybe you're thinking of the banishment of infinitesimals?

Starting at "But a certain reservation" and ending where?

Merriam online has definientia with 'series' and 'enumeration'. (There's another definiens closer to your sense, but I don't think it's common, and especially I've never seen 'list' to mean 'set' in this kind of mathematics or logic.) Not a nit; 'set' (acutually 'subset of the domain') is the word to use.

second reading — Banno

Thank you very much for that, and for saying it. Refreshing to read something like that in this forum.

"S" is true IFF X

And we can't just substitute any sentence p for S and X. — Banno

'S' is true iff X

is the general definitional form for a predicate symbol, whether 'is true' or other.

Then Tarski wanted to specify what X should be for the definition of 'is true'.

He came up with S.

I don't see opacity or circularity.

I chose "list" by way of avoiding using a technical term — Banno

'set' is no more technical than 'list'

The list is not in any particular order and might be innumerable. — Banno

I usually take 'list' as 'sequence' or 'series' or 'enumeration'.

And since we're interested in hewing to Tarski, the term to use is 'set'.

A 1-place relation symbol maps to a 1-place relation on the domain. (A 1-place relation on the domain is a subset of the domain.)

if the difficulties of opacity and circularity can be overcome. — Banno

What difficulties of opacity and circulaity?

I don't know what sense of 'opacity' you have in mind.

And Tarski's formulation is not circular. Indeed he stated a requirement that a definition not be circular, and he gave one that is not circular.

The thing about the first is, is it incontrovertible. — Banno

Right.

But recall that my unpacking was a conditional:

If 'snow' stands for blahblahblah and 'white' stands for 'bleepbleepbleep', then

'snow is white' is true if and only if blahblahblah is bleepbleepbleep.

That's a consequence of the Tarski formulation but not an equivalent (since the Tarski formulation doesn't have antecedents like that).

That is the basic idea.

For a 0-place function symbol, the denotation is a member of the domain.

For an n-place (n>0) function symbol, the denotation is an n-place total function on the domain.

For a 0-place relation symbol, the denotation is a truth value.

For an n-place (n>0) relation symbol, the denotation is an n-place relation on the domain.

list — Banno

I prefer 'set' rather than 'list', since 'list' could be taken as a sequence of the things in the set, or even suggesting a countable sequence.

item and the list is closer to the strategy of designation and satisfaction Tarski adopts. — Banno

That's an excellent point.

I like the second because it's illustrative. And I like the third because, as you observe, it is getting closer to the actual formal method (and also hews closer to Tarski's "plugging in" strategy).

the denotation is not the expression but the items themselves. — Banno

Of course the denotation is not the expression. The denotation is, formally, as given by the method of models.

Consider
"The kettle is boiling" is true if [and only if] the kettle is boiling[.]
"The kettle is boiling" is true if [and only if] the water in the kettle has reached the temperature at which its vapour pressure is equal to the pressure of the gas above it.
"The kettle is boiling" is true if [and only if] the kettle is one of the items in the list of things which are boiling.
The last strikes me as most closely resembling what Tarski does. — Banno

I don't know why you regard it as the most close. All three seem reasonable to me. Though the first is Tarski's own form.

The advantage of Tarki's form is that is is general. It applies to all sentences.

'P' is true if and only if P.

Then we plug we plug in any sentence for 'P'.

I don't know where you're headed with this, but in case my hunch is right, I would say:

Tarski is not saying how we know that 'snow is white' is true. He's only saying what it is for 'snow is white' to be true. The latter may help for the former. But the definition itself is only of the latter.

I unpacked 'snow is white' with the longer phrases. That works okay, because the context is extensional, not intensional. If we go to an intensional context:

Bob knows that ('snow is white' is true if and only if precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F has the achromatic object color of greatest lightness characteristically perceived to belong to objects that reflect diffusely nearly all incident energy throughout the visible spectrum)

then of course, that could be false.

The motivation for the unpacking was just to show that a careless reading of Tarski's formulation as being vacuous for being tautological would be not only prima facie incorrect but illustrated as incorrect.

Tarski's definitIon itself:

For any sentence 'P':

'P' is true if and only if P.

An instance:

Let 'P' be 'snow is white'.

'snow is white' is true if and only if snow is white.

They have the same denotation and extension (but not the same intension).

There's perhaps a slight problem with the choice of 'snow' for the Tarski example.

'snow' in the sentence is a noun, not an adjective.

But 'snow' is a mass noun, so it's not as easy to work with here.

Better might have been:

The snow on the lawn is white.

Or maybe 'cueball':

'The cueball on the table is white' is true if and only if the cueball on the table is white.

/

is it really the case that the former is the denotation of the latter? — Banno

No.

the phrase 'precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C)' does not denote the word 'snow'

and

the word 'snow' does not denote the phrase 'precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C)'.

Or do they just happen to denote the very same things, the denotation being those very things? — Banno

They denote the same thing.

we don't want it to be the. case that one doesn't know what snow is until one knows it is precipitation in the form of small white ice crystals formed directly from the water vapour of the air at a temperature of less than (0°C). — Banno

That's a separate epistemic question. But I don't see how it's a problem for Tarski's definition or my remarks about it.

Indeed, the second is a formal way of saying the first.

But necessity and sufficiency (the biconditional) does not enter there, in the semantics, but rather in the definitions, which are syntactical.

(Note that Tarski's definition of 'is true' is also in canonical biconditional form.)

For uniformity of style we could say 'relation symbol' rather than 'predicate symbol'.

Then have:

The model maps n-place (n any natural number) relation symbols of the language to n-place relations on the domain, and maps n-place (n any natural number) function symbols to n-place functions on the domain.

I think we could say that the extension of a relation or function symbol is the relation or function the symbol maps to.

I just now added:

I think we could say that the extension of a predicate or function symbol is the relation or function the symbol maps to. (?)

The denotation of a word is the thing the word refers to. In formal semantics, the model maps n-place (n any natural number) predicate symbols of the language to n-place relations on the domain, and maps n-place (n any natural number) function symbols to n-place functions on the domain.

That is semantical.

I think we could say that the extension of a predicate or function symbol is the relation or function the symbol maps to. (?)

/

Where biconditionals (necessity and sufficiency) enter are in definitions:

Definitional Axiom Schema - predicate symbol:
Px1...xn <-> Q
where P is a new n-place predicate symbol; x1,..., xn are distinct variables; and Q is a formula (of the language of the source theory) in which P does not occur and in which the only free variables are among x1,..., xn.
(If n = 0, then P is just a propositional symbol and there are no free variables in Q.)

Definitional Axiom Schema - function symbol:
fx1...xn = y <-> Q
where f is a new n-place function symbol; and x1,..., xn, y are distinct variables; and Q is a formula (of the language of the source theory) in which f does not occur and in which the only free variables are among x1,..., xn, y; and all closures of E!yQ are theorems of the source theory.
(If n = 0, then f is just a 0-place function symbol and there are no free variables in Q other than y.)

Those are syntactical.

/

The extension of a property is the set of all things that have the property.

That is philosophical.

where exactly is "snow" denoted as snow and "white" denoted as white — RussellA

'snow' is not denoted as snow, and 'white' is not denoted as white.

'snow' denotes snow, and 'white' denotes white.

waiting to be discovered — RussellA

Denotations are stipulated. Though it is not as clear cut in natural languages as with semantics for formal languages.

However, if given fire is hot as the expression on the right hand side, this means that the denotation of "x" as fire and the denotation of "y" as hot are already within the expression fire is hot, waiting to be unpacked.

If that is the case, then what are "x" and "y" ? — RussellA

Do you mean the denotation of 'fire' is x and the denotation of 'hot' is y?

With 'snow' and 'white' I just looked in a dictionary.

What, no leminscate to go with that?

Just to let you know that I haven't disregarded your post. I wish to give it more thought. I hope eventually to reply.

To say that I have other pursuits, closer to my heart, that I sometimes neglect for posting, does not at all entail that I think that there are not posters worth reading. I could even say that the ghost of Kurt Godel himself visits me each night and wants to give me free lessons, but I can't take him up on it, because mathematics is far from my main pursuit. That wouldn't entail that Kurt Godel is not worthy of me! — TonesInDeepFreeze

Too bad I don't have an emoticon to express that for you.

Forumcombing is itself a distraction that I probably shouldn't allow myself as it eats so terribly into my time needed for my main pursuits.
— TonesInDeepFreeze

Of course. We are not worthy of you. — apokrisis

You did it again! You falsely twisted what I said to reflect it in the worst possible way, little doubt as a spite you're exercising. You are incorrigible.

To say that I have other pursuits, closer to my heart, that I sometimes neglect for posting, does not at all entail that I think that there are not posters worth reading. I could even say that the ghost of Kurt Godel himself visits me each night and wants to give me free lessons, but I can't take him up on it, because mathematics is far from my main pursuit. That wouldn't entail that Kurt Godel is not worthy of me! Get a grip, man.

lacking a prime characteristic of the true crank — apokrisis

He's a variation.

And yet you only hammer harder when that someone finds you being overly hostile and wants to laugh you off? — apokrisis

I have recently.

Your annoyance with me should not permit you to read into my plain words things that are not in them, not even plausibly, not to willfully misconstrue what I say in the worst way, and not to strawman me over and over as you did yesterday. That is beneath you.
— TonesInDeepFreeze

You wildly exaggerate. — apokrisis

Not this time.

My impression is that we do not prove sqrt(2) is a number, but instead we assume it is a number by means of the completeness axiom. — keystone

(1) In set theory, there is no completeness axiom. Rather, we prove as a theorem that the system of reals is a complete ordered field.

(2) We assume axioms. (Or, in another view, we don't even assume them but rather merely investigate what their consequences are.) The theorems we derive from the axioms are not "assuming by means of". Okay, in a broad loose way of speaking, someone might say that the theorems are essentially just "assumptions" unpacked from the axioms. But that really muddies the matter terribly. Granted, everything we prove is, in a sense, "already in the axioms", but that obscures:

Yes, often we adopt axioms to prove the theorems we already know we want to have. But so what? That is, as they say, a feature not a bug of the axiomatic method.

And any alternative mathematics that is axiomatized is itself going to have that feature. So there's no credit in faulting set theory in particular for that.

And yes one might want for the axioms to be intuitively correct ("true") even if the theorems might be surprising. And with set theory, people's mileages vary. I find the axioms of set theory to be exemplary in sticking to only principles that are in concordance with the intuitive notion of 'sets'.

(3) So getting back to my earlier point: We prove that there is a unique positive real number r such r^2 = 2, and then we prove that r is not the ratio of two integers. Not the other way around as, if I recall, you suggested.

Now you are on a site where you get the chance to learn! — apokrisis

I glean a thing or two here and there. But posting is only a side hobby. I don't have ambitions for philosophy. Sometimes, though, I see things in discussions that I can't resist finding out about more, then I look them up or grab a book.

A scientific mindset means making the creative leap of forming a hypothesis, properly deducing the general constraints of that hypothesis, then inductively confirming the truth or otherwise of that hypothesis in terms of the observed particulars or practical consequences. — apokrisis

Of course.

If you are strong in one of the three aspects of reasoning, why would you be content with leaving the other two weak? — apokrisis

I don't know that I am so terribly relatively weak. And I'm never content. But there's so much else I also need to be doing. Forumcombing is itself a distraction that I probably shouldn't allow myself as it eats so terribly into my time needed for my main pursuits.

So when you hammer on posters, some are indeed just fools or cranks. But also, they might at least be discovering something about how to abductively form hypotheses, or inductively confirm their theories. — apokrisis

That's disastrously overgenerous. I've studied cranks for over 20 years, in forums from here to Timbuktu, and (speaking of inductive inference) one thing is clear: They never learn. They are dogmatic, irrational, intellectually dishonest, and narcissistically self-sure to the core. They persist in their favorite forum for years spewing confusion and disinformation.

reasonable person — apokrisis

I am reasonable in forums. Almost always, my first posts to a crank are without attitude. Merely a statement of the correction. Then, over time, the crank entrenches with even more dishonesty and often with passive put downs and things like that. Eventually, what becomes salient is the crank himself. And eventually I frankly say what is up with them. Believe me, I have so many times practiced restraint hoping that a crank might, miraculously, come to reason. Never happens. And that is not a function of my style. Hundreds and hundreds of other posters in many forums have tried with cranks, and they always fail to get anywhere with it. Not even a millimeter. Always.*

In this thread, you're seeing only recent interchanges. But there is a context with this poster going back over a year(?) or two years(?). I don't know whether you've read much of those threads. If not, then I would understand that you think my approach is arbitrarily harsh. (By the way, this poster is not as overtly dogmatic as usual cranks. Indeed his skill is to deflect by feigning that he is considering the corrections, which I perceive to be disingenuous.)

/

Your comments in your above post are well taken though. Even if I am in countering mode in this post, probably soon later I'll reflect more and benefit more from your point of view.

But now that you've given me advice, I will return the favor:

Your annoyance with me should not permit you to read into my plain words things that are not in them, not even plausibly, not to willfully misconstrue what I say in the worst way, and not to strawman me over and over as you did yesterday. That is beneath you.

/

* Except that one fellow I wrote about recently. The sole counterexample.

Have you noticed how much you assert the negative so as to avoid having to support the positive? — apokrisis

I don't usually support the affirmative because I'm humble enough to admit that I don't have the vision, education, confidence and constancy to arrive at a fixed philosophy. If I were a philosopher, I'd be nowhere in that way. But I'm not a philosopher.

What is life without some form of ontological commitment? — apokrisis

That is a great line.

You yourself have said you have no philosophy to defend on this forum, just a self-appointed need to police it for its mathematical thoughtcrimes and disinformation campaigns. — apokrisis

Having no philosophy is not a disqualifier. Posting is not paintball where you can't participate unless you you are on one of the teams. Not having a philosophy doesn't entail that one doesn't have meaningful things to say. And I find it refreshing when a person doesn't have a philosophical ax to grind.

There are no thought crimes. On the contrary for me. As I don't hew to a particular philosophy, I don't have strong oppositions to other philosophies; and I relish that there are so many tantalizingly different philosophies of mathematics and formal systems; and I believe that freedom to imagine is to be cherished. Spewing of disinformation though is abundant. Moreover, much of my posting is not just making corrections. Your categorical reduction is false. And beneath you, just as your multiple strawmans earlier

I've seen that cartoon, and it is funny.

Of course not. It's not even a formal claim.
— TonesInDeepFreeze

I was teasing. — apokrisis

Okay, you were joking with the 'formal' part. Maybe because you perceive me as asking posters to back up with formal proofs? Or you think I can be charactured that way? I don't know. Anyway, of course I don't ask people to provide formal proofs of informal assertions.

If you gatecrash a comment, you could at least have the courtesy to set out your reasons for your assertions. — apokrisis

Oh come on. I didn't "gatecrash" anything. You posted essentially a one-liner on the subject, itself not an argument. That's fine. And it should be allowed that one may reply in kind. And even if a poster replies tersely to a longer argument, that's not "gatecrashing" or necessarily even rude or whatever.

So you reveal yourself as a pragmatist. — apokrisis

I don't have a philosophy of mathematics; and not one that could be called anything, including 'pragmatism'.

Infinity is a useful idea as far as it goes in the real world of doing things — apokrisis

I am sympathetic to that idea. But I don't personally stake my own understanding of infinitistic mathematics primarily to it.

Truncating pi is practical. — apokrisis

Of course no one expects engineers to write an infinite sequence of digits.