Let me explain it clearly then, since you seem to be having trouble understanding. — Metaphysician Undercover

That's condescension coming from a person who can least afford it.

argues a philosophy — Metaphysician Undercover

I haven't argued a philosophy.

Many principles employed in modern mathematics, axioms, have not been empirically proven. — Metaphysician Undercover

It's dogmatic of you to preclude that interest in abstract mathematics must be dogmatism.

And I have not claimed that abstract mathematics has the kind of direct empirical correspondence that you dogmatically require. However, I do observe that it is used for, and has been a crucible for, the sciences and for the very technology you are using to be a condescending boor.

Moreover, whatever one's regard for mathematics, it is not dogmatism to point out what its actual formulations are as opposed to dogmatic attacking ignorance and misconstrual, such as yours, of the formulations.

labeling those who doubt these unproven principles as cranks — Metaphysician Undercover

I have never faulted anyone for doubts about axioms or abstract mathematics. Indeed, the literature of debate regarding doubts and criticisms of various mathematical approaches fascinates and excites me and has my admiration. What I have done though is point out when people blindly attack mathematics from ignorance, confusion, stubbornness and dogmatism. There is a Grand Canyon of difference between, on one hand, doubts and reasoned critique and, on the other hand, attacks from willful ignorance, frothing confusion, and sophomoric dogmatism.

I am curious how you got so mixed up here:

Which axiom do you claim is false?
— TonesInDeepFreeze

The above which I have underlined.
— Philosopher19

What? The underlined passage is not an axiom of set theory. And it's your claim, not mine, so there's no reason for me to defend it. You are extremely confused. — TonesInDeepFreeze

I have no interest in trying to accommodate you any further. — Philosopher19

I don't seek accommodation from you. Cleaning up your notation would be a favor to yourself.

I think you have failed to prove your position — Philosopher19

My main point is that ExAy yex is inconsistent in set theory. I proved it.

You made your only progress for yourself thus far when you replied that, given such a proof, you think the axiom used is false.

we'll have to agree to disagree — Philosopher19

As I asked before, what is the operative meaning of that? Your reply was the circlarity that 'agree to disagree' means to agree to disagree.

Essentially, I was looking for a reply to:

Call the set of all sets X. Call any set that is not X, a Y. X contains all Ys plus itself. Every set Y is a member of X. Show me how this is contradictory. — Philosopher19

I showed that the very first sentence is inconsistent with subsets.

Which axiom do you claim is false?
— TonesInDeepFreeze

The above which I have underlined. — Philosopher19

What? The underlined passage is not an axiom of set theory. And it's your claim, not mine, so there's no reason for me to defend it. You are extremely confused.

Then there is an issue in the manner in which you take subsets. — Philosopher19

Finally we're making progress. Yes, if you require that the theory uphold "There is a set of all sets" then you must reject the axiom schema of separation. And you are welcome to propose an alternative (otherwise, good luck doing any set theory without subsets).

And I've read your 'V' stuff before, and I commented on it with exactness. But if you wish to engage me with the additions you've made now, then, to start, you need to clean up the incoherent notation. In a previous post, I suggested how you could do that, but you ignored.

I suggest you parlay the progress you made when you identified that it is the subset axiom (axiom schema of separation) that is the root of your disagreement with set theory. But you should also check out the rest of the axioms to see which you might also reject.

Do you not see that you have hurled insults at me accusing me of not giving you proof — Philosopher19

(1) Pointing out that you have not proved something is not an insult.

(2) You began the volley regarding 'dogmatism' as you claimed that mathematicians are dogmatic (arguably a general insult).

(3) Whatever insults I might post do not make me dogmatic.

suggesting that you gave me proof and then provide a link to something that you said to someone? — Philosopher19

I didn't say that I addressed a proof to you personally. You said that I had not proved my claim. So I correctly said that is false and I gave you (and whomever else is reading) a link to a post in which I did prove the claim.

I checked the link (in an attempt to be charitable) — Philosopher19

It's no charity to me. It's your own improvement that would be gained by understanding the proof.

I am rejecting the rejection of a set of all sets. — Philosopher19

And I proved that there is no set of all sets, using only an instance of the axiom schema of separation.

It is in the definition of the semantic of "set" — Philosopher19

It is not unreasonable to desire a set theory that upholds our everyday notion of 'set'. However, certain difficulties arise. For example, our everyday notions might include that for every property there is the set of all and only those sets having that property. But that doesn't work, as Russell's paradox reveals.

And people also like to have a theory grounded axiomatically. Trying to have both - our everyday notions and axiomatization - presents more difficulties.

So, for those inclined to axiomatics, we have to admit that not all everyday notions will be preserved. But that is not contradiction; rather it is modesty.

an axiom I showed as being false — Philosopher19

Which axiom do you claim is false?

Call the set of all sets X. Call any set that is not X, a Y. X contains all Ys plus itself. Every set Y is a member of X. Show me how this is contradictory. — Philosopher19

As long as we have taking of subsets, the inconsistency comes with the assumption that there is a set of all sets. I proved that.

I will repeat the last part of the beginning of this post again: It is contradictory to say you can have more than one X, but there's no such thing as a set of all Xs. — Philosopher19

In context of the mathematics you are denouncing, a contradiction is a sentence and its negation. If you cannot show that set theory implies both a sentence and its negation , then you have not shown that set theory is inconsistent.

You may have your own notion of 'contradiction'. Your own notion of 'contradiction' may be that something is to you incompatible or counterintuitive or not in accord with everyday notions. And if that is the definition of 'contradiction' that we use, then, of course, I cannot deny that set theory has results that you find incompatible or counterintuitive or not in accord with everyday notions, so under that notion of 'contradiction', set theory is contradictory. But, again, that is not what mathematicians mean by 'contradiction'. And as 'contradiction' is regarded in mathematics as a sentence and its negation, you have not shown set theory to be contradictory.

"you can have more than one X"

Exy ~x=y

"there's no such thing as a set of all Xs"

~EyAx xey

Observe that Exy ~x=y and ~EyAx xey is not a statement and its negation, nor have you shown how they imply both a statement and its negation.

I have addressed the "axiom" which you present as an objection to the set of all sets. — Philosopher19

The axioms I used is an instance of the axiom schema of separation. You have not addressed that.

Also, you putting 'axiom' in scare quotes is silly and jejune.

Again, you are defending a contradiction. — Philosopher19

AGAIN, you have not shown that "there does not exist a set of which all sets are a member" is contradictory. Au contraire, I have shown a proof that "there does exist a set of which all sets are a member" is inconsistent with taking subsets.

Consider that it is you who is being dogmatic and not me. — Philosopher19

I considered it:

(1) I refer to axioms and inference rules such that it is objectively, publicly, mechanically verifiable whether a given sequence is or is not a proof from the axioms with the rules. I show certain proofs with those axioms and inference rules.

(2) I use standard definitions in mathematical logic. I point out when criticisms of mathematical logic are based in misunderstandings of the definitions. But I allow that other people may have offer different definitions, though they should be clear, non-circular, and properly formulated, and as long as the context is made clear and there is not confusion by mixing contexts with conflicting definitions.

(3) I happily embrace that other people may propose a wide range of different axioms and rules and show proofs relative to those axioms and rules. These include, among others, constructivist and intuitionist, finitist, strict finitist, predicativist, multi-valued, relevance logic, free logic, and even para-consistency. Even ersatz proposals and even non-axiomatic sketches, though I may criticize if they are not coherent or are presented with ignorant, confused, and incorrect criticism of abiding mathematics.

(4) I don't opine whether mathematics is or is not to be regarded merely as application of axioms and inference rules. And without commitment to a philosophical stance, I do countenance considerations in a wide range of alternatives to mere extremist formalism (which itself I don't opine to be necessarily incorrect), including truth regarded in different ways such as reasonable formalism, intuitiveness, correspondence, coherence, realist, structuralist, fictionalist, consequentialist, contextualist, operationalist, pragmatist. constructivist and intuitionist, common sense everyday notions, and I even allow the legitimacy of interest in para-consistency, and even (brainstorming) contrarianism. So not only am I not dogmatic, but to the contrary, I am liable to criticism for being too agnostic and. lacking philosophical commitment, too philosophically timid.

(5) I happily admit that set theory and mathematical logic themselves have certain difficulties (call them even 'mysteries') that I can't completely explain.

(6) I happily admit that my knowledge of set theory, mathematical logic, mathematics, philosophy of mathematics, and philosophy only extends to some basics and that I am not an expert. And I welcome being corrected on anything I've posted that is indeed incorrect, and am happy to post recognition of the correction and to retract as needed. And, when I have myself noticed that I made mistaken remarks or claims that I realize are not on solid footing, then I post a correction.

You:

(1) Keep demanding that your position is correct and that mathematics is incorrect, by repeating your argument while ignoring the specific points demonstrating the errors in your argument.

(2) Tendentiously make incorrect denunciations of mathematics and mathematicians that you don't know anything about.

(3) Accuse non-dogmatists of dogmatism while not facing your own dogmatism.

"primes" indicates a relation to each other. — Metaphysician Undercover

'is prime' is a predicate, not an ordering.

What is the set then? — Metaphysician Undercover

It is the unique object whose members are all and only those specified by the set's definition.

{0 1} is the unique set whose members are all and only 0 and 1.

the law of identity is an important law to uphold — Metaphysician Undercover

Set theory adheres to the laws of identity.

The sun, earth, and moon, as three unique points, have an order inherent to them — Metaphysician Undercover

What is that "inherent order"? On what basis would you say one of these is the "inherent" order but not the others?:

<Bob Sue Tom>

<Bob Tom Sue>

<Sue Bob Tom>

<Sue Tom Bob>

<Tom Bob Sue>

<Tom Sue Bob>

There are no spatial-temporal parameters for you to reference, just 3 people, with 3!=6 strict linear orderings. Order them temporally by birth date? Temporally by the day you first met them? Spatially east to west? Spatially west to east? Which is "THE inherent order" such that the other orders are not "THE inherent order"?

For a finite set S of cardinality n, there are n! strict linear ordering of S. What is the general definition of "THE inherent ordering" among n! orderings?

it makes no sense whatsoever to assume something without any order, — Metaphysician Undercover

We don't assume that sets don't have orderings. Indeed, for a set S with cardinality n, there are n! strict linear orderings of S. But when you say that one of them is "THE inherent ordering" then that requires saying what "THE inherent ordering" means.

"primes" indicates a relation to each other. — Metaphysician Undercover

'is prime' is a predicate, not an ordering.

[a mathematical set] consists of points, or some similar type of thing which have had all principles for ordering removed from them, therefore these points (or whatever) have no inherent order. — Metaphysician Undercover

Ordering was not "removed". There are n! orderings of the set. Remarking that you have not defined "THE inherent ordering" is not "removing" anything.

By what means do you say that there is a possibility for ordering them? — Metaphysician Undercover

There is not just a possibility. There exist n! orderings of the set. We prove that from axioms.

They have no spatial-temporal separation, therefore no means for distinguishing one from the other — Metaphysician Undercover

Members of mathematical sets are distinguished by properties other than spatial-temporal. And members of sets in non-mathematical contexts may be distinguished by means other than spatial-temporal. The queen of hearts is distinguished from the ace of spades, without having to refer to their positions in time and space.

abstraction has removed any possibility of order, so to speak of possible orders now is contradiction. — Metaphysician Undercover

Not just "possibility" but existence.

S = {queen-hearts ace-spades} = {ace-spades queen-hearts}

Abstraction has not "removed" orderings. S has two orderings:

{<queen-hearts ace-spades>}

and

{<ace-spades queen-hearts>}

Neither is "THE inherent ordering", unless you first give a definition of "THE inherent ordering".

When you reject such, and insist on the other, it's dogmaticism. — Metaphysician Undercover

To what do 'such' and 'other' refer?

That is not a proof. There are a few problems with it, but most glaring:

There is no set D such that for all y, y is a member of D if and only if y is not a member of itself. So, it makes no sense to say that S "chokes" on D (whatever a clear definition of 'chokes' might be) since there is no such D for S to choke on.

By the axioms, there is no set x such that every set y is a member of x. [...]
— TonesInDeepFreeze

You have not proven this. You have just stated it. — Philosopher19

That is false.

https://thephilosophyforum.com/discussion/comment/546798

https://thephilosophyforum.com/discussion/comment/546980

Okay, I admit, that does read like a 'Goofus and Gallant'.

Repeating the same ignorant falsehood doesn't work in mathematics. Only in politics. — fishfry

That should become a classic. Perfectly said.

I hate all those Vienna Circle types. That’s one of the reasons I’m going to keep out of these discussions. — Wayfarer

Whatever the merits or demerits of Carnap's views on metaphysics, the quote I mentioned does have wisdom.

And one may have one's own reasons for eschewing a conversation, but having a dislike of certain philosophers is not much of a rational basis for rejecting a conversation about mathematical logic.

The crank claims that mathematics is wrong. Not just that he proposes different mathematical conventions and definitions, but that the more ordinary conventions and definitions are blatantly wrong onto themselves. The crank cannot understand the stipulative nature of definitions. And the crank claims to find contradiction in mathematics when the crank has only found things that are not actual contradictions but instead are things that happen to be counterintuitive to him. The crank uses sophistry, evasion, raw repetition and ignorance for his position that only his own conception is right and that mathematics is wrong. The crank fancies that he eviscerates mathematics though he does not know even the least of its basics and horribly misconstrues the few bits that he has happened to come across. Thus, the preponderance of the crank's attack is the strawman. The crank is not interested in learning about the subject on which he so tendentiously opines. Instead, he is only interested in announcing his personal truths from the soapbox. The crank never (or virtually never) admits a mistake. All of that is dogmatism.


The logician says that from certain conventions, axioms, rules, and definitions, certain things follow and certain things do not follow. And the logician allows that people may set up different conventions, axioms, rules, and definitions. And the logician might even allow that proposed frameworks may have value even though they have not yet been axiomatized. The logician admits that definitions are stipulative so that definitions themselves are not inherently true, and that we may regard enquiries that proceed with different definitions. The logician seeks scrutiny of his work and is always eager to correct any errors found in his formulations. The logician admits that certain questions are not answered and that there is much still unknown. All of that is the antithesis of dogmatism.

You claim that quote is dogmatic. What is your non-dogmatic basis for that claim?

The quote is not dogmatic, I say non-dogmatically. The quote describes the way mathematical logic uses certain terminology and certain other plain facts about mathematical logic. It is apropos to mention those terminological conventions and basics of mathematical logic, since the context of the discussion is Godel's theorem, which is a subject in mathematical logic. Meanwhile, I allow that anyone is welcome to stipulate their own terminology and even to propose an entirely different framework for consideration of mathematics. However such a proposal would be subject to the same scrutiny for coherence and rigor to which mathematical logic is subject. That is it the antithesis of dogmatism.

"It is not our business to set up prohibitions, but rather to arrive at conventions." - Rudolph Carnap

What dogmatism do you think you have witnessed?

I'm talking about sentences in the language of arithmetic. I don't know whether these matters bear upon your areas of mathematics.

I am pretty rusty on this stuff, so take this modulo a grain of salt:

The Godel sentence G "says":

"For every n, it is not the case that n is the Godel-number of a proof of the sentence with Godel-number g."

G is a sentence purely in the language of arithmetic. The "it says" about proofs and Godel-numbers is seen and proven (in the meta-theory) with regard to the construction of G per the arithmetization of syntax.

And, G has Godel-number g.

The part "it is not the case that n is the Godel-number of a proof of the sentence with Godel-number g" is a computable property. Let's call it 'C'. So G is of the form:

For all n, Cn.

Now, for concision, let's say we're looking at some particular system S.

Godel-Rosser proves "If S is consistent, then both G and ~G are not theorems of S."

And let's say that by 'true' we mean true in the standard model for the language of arithmetic. Godel did not himself have formal model theory to reference, but in context we may say that his context might as well be tantamount to it. Moreover, we could dispense the formality of models by just agreeing that 'true' means what it ordinarily means to mathematicians who don't care about mathematical logic. For example, '0+0=0' is simply true and '0=1' is simply false.

So, either G is true or ~G is true. So, on that basis alone, we know that there is a true sentence that S does not prove. But that is not constructive - it uses excluded middle and doesn't tell us specifically which one of the two is the true one.

But we can constructively (I think?) show "If S is consistent then G is true" anyway.

Now, let's look at a certain kind of arithmetical sentence. These are sentences in the language of arithmetic all of whose quantifiers are bounded. For example:

"For all n<20, if n is prime then n has a twin prime."

For such sentences, there is an algorithm to decide their truth. Moreover, it is said that from the sentence itself, we can "read off" the algorithm (please don't ask me the technical definition of "read off" - I have not yet pursued how to formalize it).

Now, where I tripped myself up earlier in this thread is that I might have conflated the fact in the above paragraph with the fact that we do easily prove "if S is consistent then G is true", as I am not clear whether that proof is one that also is "read off" from the sentence itself, in context of the construction of the sentence vis-a-vis the arithmetization of syntax.

Modulo any typos or formatting glitches from one forum to another, of course, I agree that the function is partial.

And I take the poster's word about the growth rate.

But I don't know enough about the busy beaver problem and its comparison with incompleteness to say anything about it.

Suggesting the paradox is an artifact of language and no real part of set theory. — tim wood

If you're referring to Russell's paradox, it is a matter of logic and is not peculiar to set theory, but rather applies to any 2-place relation R:

For any 2-place relation R, there is no x such that, for all y, y bears relation R to x if and only if y does not bear relation R to itself. (With set theory, R happens to be 'is a member of'.)

We're addressing one particular metric: length of of the sum of the lengths of the formulas. This is not meant in itself to imply anything about the time it takes human beings to devise or compose proofs. This is not paradoxical.

But if we adhere to strict "truth=provability" principle, then the sentence is not true even in the metatheory, if it assigns truth to sentences subject to their provability in the object theory. — SophistiCat

What "truth=provability" principle do you have in mind? What is its mathematical formulation? Meanwhile, the incompleteness theorem proves that the set of provable sentences does not equal the set of true sentences. [Often now, I'll leave tacit the usual qualifiers such as "provable in system S" and "true in the standard model".]

Anti-realists recognize arithmetical statements as true relative to particular mathematical theories — SophistiCat

True relative to models of theories. (Of course though, if P is a theorem of a consistent theory S then P is true in any model of S.)

which are as fictitious as any other such theories — SophistiCat

Not necessarily for arithmetical theories or even the arithmetical part of broader theories.

Keeping in mind that our (eastern) set has been scrubbed and disinfected of self-contradictory sets? — tim wood

A sentence (such as set existence assertion) is found to be contradictory with other sentences by being found to be a self-contradiction (logically false) or by being found to contradict previously proved sentences. So what the demons keeps or throws out, will be based on what he's already kept.

But set theory can't do that and be recursively axiomatized, because there is no algorithm to determine whether a given sentence contradicts a previous set of statements. That's why I say the demon is magical, because he's able to make immediate determinations that are not calculable even in principle.

It is true that there are maximally consistent sets of sentences. But membership in such sets is not algorithmically decidable.

his naive inchoate practices approximate those of logicians c. 1920. — tim wood

Which logicians and what formulations do you have in mind?

Nope, if have taking of subsets, but then stipulate that we are not allowing in particular a set of all sets, then we could still derive a contradiction.
— TonesInDeepFreeze
And this I do not see. — tim wood

I later corrected my typo of omission there.

It should be:

If we have taking of subsets, but then stipulate that we are not allowing in particular a set of all sets that are not members of themselves, then we could still derive a contradiction.

Do you want the proof?

It seems as I read it that you derive a contradiction from the idea of subsets in themselves. Am I misreading? — tim wood

Yes, you are. The axiom schema of separation is not inconsistent itself. It is inconsistent with the claim that there is a set of all sets.

https://thephilosophyforum.com/discussion/comment/546862

I had a typo of omission there. I fixed it now.

And I said that even if the demon throws out the set of all sets that are not members of themselves, then there would still be a contradiction, if we allow subsets. I should add that what happens is that we get an infinite process of forming subsets and then the demon throwing them out ad infinitum. But then that is not analogous to set theory, even if we grant a premise of infinite process.

I can't imagine you'd have issues so severe that you couldn't see that self-membership does not imply impossibility of being in another set.

K inside the prison {K} is equal to (is the same as) K outside the prison. — TheMadFool

Set theory does not have a predicate "inside braces".

But on our construction, the demon already sniffed that out and left it in the west as not a set. — tim wood

Then the demon is not allowing the subset operation. So the collection would not be one recognizable as serving an ordinary set theoretic role. But you do continue to say we'll disallow certain subsets:

But for the rest, there can be a set of all the other sets? — tim wood

Nope, if have taking of subsets, but then stipulate that we are not allowing in particular a set of all sets that are not members of themselves, then we could still derive a contradiction from "the set of all sets except the set of all sets that are not members of themselves".

Moreover, this demon would not be making his decisiond by algorithmic determination; his sniffing is purely magical, unlike set theory.

No. — TheMadFool

You thought about it for at least half a minute?

If there is a set of all sets, then that set has a subset that is the set of all sets that are not members of themselves, which implies a contradiction.

Thank you for your time. — TheMadFool

So you figured why this is not the case?:

without regularity
Ax({x}e{x} -> Ay(~y={x} -> ~{x}ey))

It is unprovable in the system being discussed. It is provably true in the mathematics used to discuss that system. — TonesInDeepFreeze

I should qualify that remark and others I made along the same lines.

We prove (though not in the object system) that the Godel-sentence is true on the assumption that the object-system is consistent.. That qualification might be regarded as implicit in my remarks, but it is best for me to make it explicit.

we are on quite firm ground "epistemologically" to say, without quibbling about formality, that the sentence is true when we can compute that it does hold. — TonesInDeepFreeze

I should put that remark on hold. I need to figure out whether saying that we have a "computation" is correct.

A set {P} that contains itself is the set that can't be a member of another set! — TheMadFool

That is so blazingly incorrect that it scorches the core of this planet.

Every several thing east is a set in itself, but the collection of them, in the east, is not a set? — tim wood

Right.

realist/Platonist — SophistiCat

When we're talking about plain arithmetical truths, I don't know why we would have to go down the road of wondering about realism. I mean, non-realists still recognize the truth of arithmetical statements.

Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.
— Raatikainen 2015

Here the statement is not said to be either true or false [...] in the context of system F the Gödel statement is neither true nor false, since it can be neither proved nor disproved in that system. — SophistiCat

Yes, the theorem itself, as you quoted it, does not mention truth. But from the theorem, we do go on to remark that the undecided sentence is true.

And the statement is neither true or false in the system on an even more fundamental basis than that it is undecided by the system:

There is an even more fundamental reason that the object-theory does not yield a determination of truth. That is that the object-language does not have a truth predicate. There's a subtle difference: A theorem of the object-theory is true in any model of the theory, so in that sense one would say that the object-theory does determine the truth of certain sentences. But the object-theory does not itself have a theorem that the sentence is true in models of the theory (or else, the object-theory would be inconsistent per Tarski's theorem). — TonesInDeepFreeze

I’m afraid to say that you’re [Pfhorrest] splitting hairs. — Wayfarer

I don't think he is. The distinction he's making is very important.

Godel's about whether there are things that are true but aren't provable. — Pfhorrest

That is exactly the most salient oversimplification that causes misunderstanding.

You know the following, but it bears emphasizing:

There is no mathematical statement that isn't provable. That is, for any mathematical statement (even a self-contradiction) there are systems that prove the statement.

Godel's theorem is that for any given system S of a certain kind there are statements F in the language for S that such that S proves neither F nor ~F.

It's a matter of quantifier order:

Godel: For any system S of a certain kind, there exist statements undecided by S.

False: There exist statements F such that for any system S of a certain kind, F is undecided

Godel only shows something about the relationship between a formal system and statements in it: that some systems can't prove some things they're capable of talking about either way, even though we can know, through in a proof made in a higher-level system, that those things are true. — Pfhorrest

That seems to me to be a reasonable summary.

Something that is assumed to be so, but can’t be proven to be so? Isn’t that what the issue is about? — Wayfarer

I don't know what issue you mean when you ask what the issue is about. But for incompleteness, it's not just a matter of having to assume things to prove things.