Where does one find a definition of 'actual' that includes 'completed'? — TonesInDeepFreeze

It might go the same way it came (per Mary Tiles). — frank

What do you mean by that? And are you referring to her book 'The Philosophy Of Set Theory'? If so, what in particular do you have in mind from that book?

In ordinary English, we use these senses:

(1) A count is an instance of counting. "Do a count of the books."

(2) A count is the result of counting. "The count of the books is five."

A number (we're talking about natural numbers in this context) is a count in sense (2). That doesn't preclude that a number is a mathematical object.

do you agree that it is necessary that there is a thing counted — Metaphysician Undercover

To have a count (in sense (1)), you need something to count. (Except in the base case, there is the empty count.)

If "1" does not refer to the book, as well as what you call the number, then there is nothing being counted — Metaphysician Undercover

Numbers also have other aspects than being counts (or 'results of counts' depending on how exactly we might define 'count').

we cannot dispense with the fact that "1" must refer to the object being counted, a book — Metaphysician Undercover

We better dispense with that notion. It's nuts. A number is not a book.

To have a true count, "1" must refer to the first book, "2" refers to the first and second together — Metaphysician Undercover

'1' does not denote a book. And 'together' is not defined by you.

The most common mathematical understanding of counting is bijection.

For a set with only one book - 'War And Peace' - in the set, the count is the greatest number in the range of the bijection between the set {'War And Peace'} and the set {1}, and that greatest number is 1.

The bijection is {<'War And Peace' 1>}.

For a set with only two books - 'War And Peace' and 'Portnoy's Complaint' - in the set, the count is the greatest number in the range of the bijection between the set {'War And Peace' and 'Portnoy's Complaint'} and the set {1 2}, and that greatest number is 2.

The bijection is {<'War And Peace' 1> <'Portnoy's Complaint' 2>}.

So the numeral does not denote a book, but rather it denotes the number that is paired to the book in the bijection (or, in everyday terms, in the pairing off procedure we call 'counting').

We pair 'War And Peace' to 1, then we pair 'Portnoy's Complaint' to 2. That's counting.

We don't say "''1' denotes 'War And Peace' and '2' denotes 'War And Peace' together with 'Portnoy's Complaint'". That's crazy.

I do not know the technical terms (syntax?, proof system?). — Trestone

Those are utterly basic to the subject. If you don't even know what proof is, then you can't very well explain whatever alternative system you have in mind.

For me a contradiction is, if the same statement is shown as true and not true. — Trestone

That's kind of okay in an everyday sense. But a contradiction is actually a formula of the form

P & ~P

(Or, more generally, a set of formulas that implies a formula of the form P & ~P.)

indirect or by contradiction — Trestone

I've informed you twice already that you are misusing that terminology. The particular proofs we are talking about in ordinary mathematics are not indirect (and 'proof by contradiction' is another term for 'indirect').

That is, layer logic disproves all formulas of the form 'P & ~P' [?'] — TonesInDeepFreeze

No, only when there are different layers used. — Trestone

So, for example, within a single layer you don't prove "It is not the case that both 2 is even and 2 is not even".

~0=1 Trestone: false in layer math — TonesInDeepFreeze

Really, you think it is false that 0 does not equal 1?

~Ex (x is a natural number & x>x) Trestone: false in layer math — TonesInDeepFreeze

Really, you think there is a natural number that is greater than itself?

In many cases it helps, that in layder 0 all sentences have truth value "undefined". — Trestone

You haven't said how, in general, one evaluates the truth of atomic sentences, compound sentences, or quantificational sentences in any layer.

Non-Turing algorithms — Trestone

Pray tell, what class of algorithms do you have in mind that are not Turing machine computable? Actually, to reduce even more confusion and misinformation than you've already posted, pray don't tell.

You begin with:

(t marks the layers, W(x,t) ist the truth value of x in layer t, -w stands for „not true“ or „false“
ther value „undefined“ I left out to make things easier).
Be M a set and P(M) its power set and F: M -> P(M) a bijection between them (in layer d)
Then the set A is defined with W(x e A, t+1) = w := if ( W(x e M,t)=w and W(x e F(x),t)=-w ) — TonesInDeepFreeze

like in the proof of Cantor, I asume herre that such a F exists. — Trestone

You are completely confused.

Cantor proves that such a bijection does not exist. You purport to prove that such a bijection does exist.

To prove that such a bijection does not exist, we may assume one does exits, then derive a contradiction, then conclude that such a bijection does not exist (and that is not indirect proof).

But to prove that such a bijection does exist, you can't start by assuming that such a bijection does exist. That is question begging.

Moreover, we don't even need to assume such a bijection exists and derive a contradiction. Rather, we argue from universal generalization: Let f be any function from S to PS. Then we show that S is not a surjection.

x is a member of a layer set and therefore itself a layer set. — Trestone

You skipped the problem I mentioned. First you use 'x' to stand for a sentence, then 'x' to stand for a set. That, like virtually everything else you say, is nonsense.

the proof about the power set can be similary be "unproofed" — TonesInDeepFreeze

"unproved" is nonsense terminology unless you specify which of these you mean:

1) A proof of ~P in a given system..

(2) A meta-proof that P is not a theorem of given system.

(3) Pointing out a line in a purported proof in a given system that it is not actually an allowed line in that system (i.e. pointing out where a purported proof is not an actual proof).

What you seem to think is that you have accomplished a variation of (1) by showing that in your own system it is not the case that there is not a bijection between S and PS. But you fail from the very start by assuming that there is such a bijection.

"You will know them by their fruits" (Matthew 7:15-20) — Trestone

For crying out loud, you compare yourself to Lenin, the hero of the Cave, Christopher Columbus, and Matthew the Apostle. Maybe get a grip and cut back on the self-aggrandizement.

Anyway, the fruitfulness of set theory is in having a recursively axiomatized theory for the mathematics of the sciences, conceptualization and formalization of computability theory that enables, among other things, the computer technology we all use to type things on the Internet, a treasure of peer-reviewed articles that provide reading about fascinating intellectual accomplishments, and a starting point for approaches that differ from classical mathematics, including multi-valued logics, constructive mathematics, paraconsistent logic, and others that can be quite exotic. Meanwhile, the fruit of your postings is from the poisoned tree of ignorance.

I am sorry that I can not answer most your questions to formal details. — Trestone

You're lacking not just all the formal details, but even a coherent outline.

F: M -> P(M) a bijection between them — Trestone

Again, you can't assume that there is bijection from M onto PM. You are purporting to prove there is such a bijection, so you can't do that by assuming there is one. Unless layer logic is so meaningless that it allows proof by question begging.

The problem is not just a lack of familiarity with the technicalities of mathematical logic, but that you don't understand even the very basics of even informal logical reasoning.

in layer math, the existence of F does not lead to a contradiction — Trestone

You don't know that unless you've proven the consistency of layer math. But, of course, you can't prove the consistency of something that is not formed with sufficient determinateness to tell what is a theorem and what is not.

hope you have a little understanding for a "Columbus" — Trestone

That would be the Columbus who brought widespread fatal disease, subjugation, and genocide. Meanwhile, you bring confusion, ignorance, and misinformation. Not as bad, so I wouldn't insist on the comparison.

Let's say that "1" refers to the number 1, which represents the count, and is also the thing counted — Metaphysician Undercover

No, let's not say that 1 is the thing counted.

The things that are counted are, in this case, the books.

'1' is a numeral.

1 is the number denoted by the numeral '1'.

1 is the count of the books.

1 is not a book.

What is the first line in each of the below proofs that is not allowed in layer math?

Show: There is no function from a set onto its power set.

Proof :

Let f be function from S to PS. Let d = {x | xeS & ~xef(x)}.

dePS.

If d is in range(f), then for some x in S we have d=f(x).

If xef(x), then ~xed, so ~xef(x).

If ~xef(x), then xed, so xef(x).

Contradiction. So d is not in the range of f. So f is not a function from S onto PS.

/

Show: ~ExAy yex.

Let Ay yex.

Let d = {x | xey & ~xex}.

If ded, then ~ded.

If ~ded, then ded.

Contradiction. So ~ExAy yex. — TonesInDeepFreeze

Note: In my previous posts, anywhere I mistakenly wrote 'level' I meant 'layer', as I guess would be obvious anyway.

In ordinary mathematical logic, contradictions are syntactical, not requiring assignment of truth values. Meanwhile, as far as I can tell, your layer logic is described primarily semantically in terms of truth values; I don't know the syntax of whatever proof system you have in mind, so I can't evaluate the means by which you would prevent (syntactical) contradictions. You could assert that provability entails soundness, but we need to prove that, not just assert it, and you can't prove it without first stating what the proof system is. — TonesInDeepFreeze

That stands without your response.

I would guess that layer logic does disprove contradictions. That is, layer logic disproves all formulas of the form 'P & ~P' (where they "reside" (or\e whatever way you say it) in the same level [should be layer]). — TonesInDeepFreeze

Is my guess correct?

And does layer math prove the following?:

~0=1

and

~Ex (x is a natural number & x>x)

And you admit that layer math does not prove the fundamental theorem of arithmetic. So layer math would not seem to offer much as a mathematical foundation anyway.

what you say is that there are three truth values and that statements are evaluated at different levels [should be layers]. You haven't given even the starting point: description of evaluation of truth and falsehood for atomic sentences, compound sentences, and quantificational sentences. — TonesInDeepFreeze

That stands without your response.

how I handle the proof of the halting problem — Trestone

You begin with:

with layer logic we have to add layers if a program has to give a value/result:
A given program halts or not in layer k for given input data. — Trestone

But no axioms or rules of inference by which to claim that.

So to follow along with you in your layer math, one just has to accept the arbitrary lines in your arguments as given by you personally (there is no objective codification). You do not provide one with a way to check whether the lines you put forth are axioms or theorems of layer math but instead one must rely solely on your dicta as to what constitutes a valid line or inference in an argument.

my earlier handling of Cantor´s diagonalization and proof in layer logic — Trestone

You begin with:

(t marks the layers, W(x,t) ist the truth value of x in layer t, -w stands for „not true“ or „false“
ther value „undefined“ I left out to make things easier).
Be M a set and P(M) its power set and F: M -> P(M) a bijection between them (in layer d)
Then the set A is defined with W(x e A, t+1) = w := if ( W(x e M,t)=w and W(x e F(x),t)=-w ) — Trestone

In ordinary logic, truth values apply to sentences. It seems that had previously been the case in your discussion of layer logic too. Here you mention the truth value of x, So I take it that x ranges over sentences there. But then we find x ranging over prospective members of the set A. So which is it? x ranges over sentences or x ranges over prospective members of sets? So far, what you've given is pseudo-math or gibberish dressed up with undefined math/logic-sounding verbiage.

Also, you mention things (which I guess are sentence) as being true or false in layers, but now here we find that functions too are things in layers. But you've not stated what a layer sis or what kinds of things can be in layers or, as I mentioned earlier, how it is determined a given atomic, compound, or quantificational sentence is true or false in a layer.

F: M -> P(M) a bijection — Trestone

Are you there asserting that there exists such an F? If you are, but without first proving the existence of such an F, it would seem to be question begging, since by supposedly refuting Cantor's theorem, you're claiming to prove that there does exist such an F.

the proof about the power set can be similary be "unproofed" like the halting problem — Trestone

Just to be clear, these are all distinct:

(1) A proof of ~P in a given system..

(2) A meta-proof that P is not a theorem of given system.

(3) Pointing out a line in a purported proof of a given system that it is not actually an allowed line in that system (i.e. pointing out where a purported proof is not an actual proof).

(4) A meta-proof that P is false in a given model of a given theory.

So, letting P = Cantor's theorem, do you you claim either (1) or (2) regarding layer math? (I take it that you do claim (4) or something like it.)

His new world is pure nonsense and fantasy for the Cave people. — Trestone

That's question begging. One can just as well say you've not left your own cave, as you are not familiar with the logic and mathematics that has been explored by generations of logicians and mathematicians who have themselves studied alternatives including types, orders, levels in set theory, quantification over theories themselves, modalities, possible world semantics, topological semantics, and even para-consistency.

all the proofs you named are valid no more — Trestone

I would guess that layer logic does disprove contradictions. That is, layer logic disproves all formulas of the form 'P & ~P' (where they "reside" (or\e whatever way you say it) in the same level).

/

How does layer logic disallow this proof (which is not indirect)?:

Show: There is no function from a set onto its power set.

Proof :

Let f be function from S to PS. Let d = {x | xeS & ~xef(x)}.

dePS.

If d is in range(f), then for some x in S we have d=f(x).

If xef(x), then ~xed, so ~xef(x).

If ~xef(x), then xed, so xef(x).

Contradiction. So d is not in the range of f. So f is not a function from S onto PS.

/

How does layer logic disallow this proof (which is not indirect)?:

Show: ~ExAy yex.

Let Ay yex.

Let d = {x | xey & ~xex}.

If ded, then ~ded.

If ~ded, then ded.

Contradiction. So ~ExAy yex.

/

Remember, I never mentioned 'truth' or 'falsehood'. I merely gave syntactical proofs. So you haven't shown how those proofs are disallowed by merely saying that truth and falsehood may alternate on levels.

As for semantics, basically, what you say is that there are three truth values and that statements are evaluated at different levels. You haven't given even the starting point: description of evaluation of truth and falsehood for atomic sentences, compound sentences, and quantificational sentences.

No, neither 'indirect proof' nor 'proof by contradiction' are correctly applied to those proofs, as they are not of the form:

Show P.
Assume ~P
Derive contradiction.
Conclude P.

by constructing the contradictions we have to use different layers,
and different truth values in different layer are not a contradiction in layer logic. — Trestone

In ordinary mathematical logic, contradictions are syntactical, not requiring assignment of truth values. Meanwhile, as far as I can tell, your layer logic is described primarily semantically in terms of truth values; I don't know the syntax of whatever proof system you have in mind, so I can't evaluate the means by which you would prevent (syntactical) contradictions. You could assert that provability entails soundness, but we need to prove that, not just assert it, and you can't prove it without first stating what the proof system is.

There is a fundamental problem with the concept of numbers. The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well. So we have to allow that "1" represents a different type of unity than "2" does, or else we'd have the contradiction of "2" representing both one and also two of the same type of unity. — Metaphysician Undercover

Now that you've somewhat clarified that, here's the best I can make of it (I don't claim to represent what you have in mind, but this is the best I can make sense of it):

1 is the count at the first member of the set, a particular unity (whatever it is). 2 is the count at the second member of the set. Etc. And '1' and '2' name different individual numbers. And 1 is the count of the members of the set with one unit. And 2 is the count of the members of a unity that is a set with two members. And a set with one member is a different kind of unity from a set with two members.

I think that's all okay. But then you conclude:

"we'd have the contradiction of "2" representing both one and also two of the same type of unity."

'2' denotes the number 2. The number 2 is the count of a set with two members. And a set of two members is itself a unity as a set. But '2' does not denote a unity; it does not denote the set that it counts. It denotes the COUNT of a set that is itself a unity. When we say that a set is a unity, we mean that it is one set, while we recognize that the number of members of the set may be greater than one.

{'War And Peace' 'Portnoy's Complaint'} is one set, which is a unity. But the count of {'War And Peace' 'Portnoy's Complaint'} is 2.

I don't see a contradiction.

near complete denial of the relation between one and unity — Metaphysician Undercover

I had said in the very post to which you now replied, "Of course the notion of 'one' is related to that of a unity." So I couldn't possibly be in denial about it. You are very confused.

Then I conclude that what needs to be discussed is clarification of P and Q. — Metaphysician Undercover

Then you need to clarify them, since you are the one who mentioned them, and you have not shown what anyone else said that is properly rendered as you did. But now you do clarify:

Count the books on the shelf for example. "Book" signifies the type of unity being counted, "1" signifies that a unity called "a book" has been identified, and a first one has been counted , "2" signifies two of these units, etc.. — Metaphysician Undercover

That might be sharpened a little, but I guess it's okay on its own. However, "The "2" represents two of those individuals together, and "3" represents three, etc" is garbled, so I asked for an example of anything said by anyone that you think is properly rendered that way. First you said that I had just read it. But, as far as I can tell, it's not in the posts of the poster you quoted. Then you said that it's actually your own notion. Then you came back around to telling me that I can find it on the Internet somewhere.

I assume you know how to use Google or some other search facility. You could simply search this if you need such a confirmation, instead of asking me to do your research for you. — Metaphysician Undercover

I didn't ask you to research for me. I just wanted to know of a particular example that YOU consider to be properly rendered by "The "2" represents two of those individuals together, and "3" represents three, etc" and "'2" and "3" represent some kind of unity". I can find all kinds of things about units and unity on the Internet. But that wouldn't tell me what in particular YOU have in mind as being properly rendered by "The "2" represents two of those individuals together, and "3" represents three, etc" and "'2" and "3" represent some kind of unity".

Here's the first entry I get when I Google that question, is 1 a prime number — Metaphysician Undercover

I said in the very post to which you now replied:

Of course the notion of 'one' is related to that of a unity. But even aside from parsing, I don't know who in particular you think holds that "The "2" represents two of those individuals together, and "3" represents three, etc". It would help if you would cite at least one particular written passage by someone that you think is properly rendered as "the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc" and "'2" and "3" represent some kind of unity". — TonesInDeepFreeze

So I made clear I wasn't asking about 1 but about the part where you write, "The "2" represents two of those individuals together, and "3" represents three, etc" and "'2" and "3" represent some kind of unity".

So your example about 1 not being a prime number does not address this.

Anyway, after first telling me that I had already read someone who said this, then saying instead that it was your own notion, then saying it is on the Internet (but citing an example not addressing it), we did get something of a better statement by you as I mentioned earlier in this post. Such long and unnecessary detours by you. As well as you flatly claimed the opposite about me when you said that I'm in "near denial" about the relation between the notions of one and unity when I had actually explicitly stated that there is a relation.

Analysis of most classical indirect proofs show — Trestone

If you have in mind the famous proofs regarding a universal set, uncountablity, incompleteness, Tarski's theorem, and the halting problem, then these have direct proofs. Any proof of the form.

Show ~P.
Assume P.
Derive contradiction.
Conclude ~P.

has a structure of direct proof.

Indirect proof is of the structure:

Show P.
Assume ~P
Derive contradiction.
Conclude P.

The non-existence of a universal set, uncountability, incompleteness, Tarski's theorem, and the halting problem do not rely on that structure.

Thus one can never say "I always lie" — maytham naei

One can say 'Everything I say is false'.

But one can't say it without self-contradiction.

(By the way, for purposes of paradox, it's clearer to refer to 'truth' and 'falsehood' rather than 'lying' and 'not lying'. Because someone can state a falsehood with lying (the person may not know that their statement is false).

In particular, you conflated lying with not telling the truth when you made the alternatives:

Lied vs told the truth.

The proper alternatives are either

Lied vs did not lie

or

Told truth vs didn't tell truth.

> "I mostly lie"

Hopefully you don't know people in your life who mostly lie. But it's still possible for someone to do so. — maytham naei

Yes, 'I mostly lie' is not self-contradictory. So what?

Hopefully you don't know people in your life who mostly lie. — maytham naei

For four years the people of the United States had the 45th president in their lives who lied as many times a day as most people breathe in and out. Probably he even lied in his sleep.

I can't make sense of your essay - from the very start where you begin by flinging around terminology used in a personal way but that you don't define nor declare as primitive.

If you are truly interested in people spending their valuable time to grasp your ideas, then you would be well served by formulating them systematically rather than merely hoping that Internet passerbys would try to figure out the meaning of your thrown-together verbiage for you.

So your argument outline is: If someone wants P, he is reminded that P contradicts Q, which he might also want, so we think about a way to adjust so that there is not a contradiction.

But neither P nor Q are stated coherently by you. And there's no reason to think anyone wants P or Q anyway.

More basically, I don't know why one would fret over any of this, since I don't know anyone who claims "the numeral "1" represents a basic unity. an individual.
— TonesInDeepFreeze

I find that very strange I hear them used that way all the time. — Metaphysician Undercover

You cut off the rest of my quote of you.

"the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc" — TonesInDeepFreeze

Of course the notion of 'one' is related to that of a unity. But even aside from parsing, I don't know who in particular you think holds that "The "2" represents two of those individuals together, and "3" represents three, etc". It would help if you would cite at least one particular written passage by someone that you think is properly rendered as "the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc" and "'2" and "3" represent some kind of unity".

You haven't really ever thought about such fundamental issues as how we use numerals, and you don't really understand why anyone else would. — Metaphysician Undercover

You don't know that I haven't thought about numerals. Actually in another thread, in posts directly with you, I posted a fair amount on the subject. And, most pointedly, I have never written that I don't understand why anyone would be interested in the subject.

I see no problem with that in itself. The problem is when we want to say that, and also that "2" and "3" represent a type of unity. — Metaphysician Undercover

I understood that; I thought you meant that you do want to take '2' and '3' as representing a type of unity, while you think that that is contradicted by 'the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc" so that it needs correction .

Am I not correct that that is your view?

More basically, I don't know why one would fret over any of this, since I don't know anyone who claims "the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc" or anyone who claims "'2" and "3" represent some kind of unity". Moreover, aside from the ill-formed English, I don't know that you're not using such terms as 'unity' and 'represents' other than in a personal idiosyncratic way. Also, I would wonder what are your rigorous mathematical or philosophical definitions of 'basic unity', 'individuals together'. In sum, I can't make sense of what you're trying to say.

Suggestion: You could reference some actual piece of mathematical or philosophical writing that you disagree with and show how you think you can correct it.

I am claiming that there is a fundamental problem with numbers. If "1", "2", "3", etc. , are used to represent unities, — Metaphysician Undercover

Yes, as I thought, you find that there is a problem with the notion (whatever it means) that 'the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc".

But (aside from even trying to parse the broken phrases) I don't know who says anything along the lines of 'the numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc". So I don't see why you think it is a problem that needs to be addressed.

There is a fundamental problem with the concept of numbers. The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well. So we have to allow that "1" represents a different type of unity than "2" does, or else we'd have the contradiction of "2" representing both one and also two of the same type of unity. — Metaphysician Undercover

I thought you meant that there is a fundamental problem with:

"The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well."

And that your supposed solution to the supposed problem is:

"[...] we have to allow that "1" represents a different type of unity than "2" does [...]"

So I was wondering who you have in mind as having said or written anywhere that:

"The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well."

Or are you saying that you yourself holds that:

"The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well."?

Or perhaps you would make clear which parts of your passage are ones you are critiquing and which parts are ones you are claiming.

Where can I actualy read anyone explaining the concept of numbers that way?
— TonesInDeepFreeze

Didn't you just read it? — Metaphysician Undercover

The post to which you responded didn't say it. So would you please link to a post or reference anything on the Internet or anywhere else?

What are the primitives of your system? What are the formation rules?

There is a fundamental problem with the concept of numbers. The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well. — Metaphysician Undercover

Whose concept is that? Where can I actualy read anyone explaining the concept of numbers that way?

as we go towards 0 and infinity, are we just supposed to say, "Oh well?" Apparently. — synthesis

What exactly do you mean by "go towards 0 and infinity"? And who apparently says "oh well" in this context?

1. Set theory does not have, in this context, formal terms 'actual infinity' or 'completed infinity'. So for formal concerns we don't need to vindicate those notions that are not even used.

2. Set theory, in this context, does not use 'infinity' as a noun, but instead uses the adjective 'infinite'. This is important since, in this context, set theory does not point to a set named 'infinity' but rather mentions that various sets have the attribute of being infinite.

3. The non-formal notion of actual infinity does not need to resort to the term 'completed'.

4. Moreover, where does one find even a non-formal definition of 'infinity' that includes 'that which can't be completed'? If 'cannot be completed' is not part of the actual definition, then it is question begging to use this to argue that 'completed infinity' is a contradiction in terms.

6. Where does one find a definition of 'actual' that includes 'completed'?

7. Cantor's work was not axiomatized by him. It was only later mathematicians who axiomatized infinitistic mathematics.

8. The set theoretic definition of 'is infinite' is given this way.

x is finite iff there is a natural number n such that there is a 1-1 correspondence between x and n.

x if infinite iff x is not finite.

9.

maths is essentially an axiomatic system, anything goes so long as you don't contradict yourself within one. — TheMadFool

Yes, in a broad sense, and from a certain point of view, that is correct:

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

Probably the particular formulation of Russell and Whitehead became less used in comparison with the arguably more straightforward approach of axiomatic set theory.

Nevertheless, interest in type theories has continued. In the literature through the decades there are many kinds of type theories and variations of type theories representing different schools of mathematical and foundational thinking.

Just to be clear, the existence of a set whose members are all and only those sets that are not members of themselves is ruled out by first order logic alone, even before we get to set theory.

For any two-place predicate F (whether F is the membership relation of any other relation) we have a theorem of first order logic:

~ExAy(Fyx <-> ~Fyy)

So set theory does not need to add axioms that prove [read 'e' as epsilon]:

~ExAy(yex <-> ~yey)

What set theory does need is to be careful not to add axioms that prove:

ExAy(yex <-> ~yey)

The diagonalization of Cantor leads to the way out:
Of course the proof is correct, but it uses the classical logic. — Trestone

Cantor's diagonal proof is intuitionistically valid.

I can't help it if your terminology is a little off the beaten path. You kept referring to a "method of models", and I couldn't even find that on google. — Metaphysician Undercover

So you opted to suggest that I'm lying about the whole thing instead of just asking "Would you please provide some links?" Another example of your jejune approach to mathematics and discussion about it.

And what I wrote:

You need to read a book or other systematic presentation of mathematical logic in which the method of models is explained step by step — TonesInDeepFreeze

[emphases added]

So you could have easily searched 'mathematical logic method of models' or just asked me. Instead you burden me with the suggestion that I'm lying about the subject because you can't be bothered to make a reasonable search. Then when you look at the links, you only skim a few parts of them, then pick a few items in them out of context and misconstrue them. And you're welcome, by the way, for the links I provided, even though your response to them is itself incorrect - including incorrectly summarizing their level, relevance, content and import.

The way to learn the subject is not by perusing articles and parts of books out of context, but rather by first starting with at an introductory level book on symbolic logic.

I tend to think it is something you made up as a ruse — Metaphysician Undercover

You tend to think irrationally or not at all.

Below are some articles and a couple of online books. For a more systematic study, if I recall, earlier this thread I mentioned particular books I most recommend.

http://page.mi.fu-berlin.de/raut/logic3/announce.pdf

https://en.wikipedia.org/wiki/Structure_(mathematical_logic)

https://plato.stanford.edu/entries/model-theory/

https://plato.stanford.edu/entries/modeltheory-fo/

http://www.math.toronto.edu/weiss/model_theory.pdf

https://en.citizendium.org/wiki/Structure_(mathematical_logic)

https://faculty.math.illinois.edu/~vddries/main.pdf

Those are concepts instrumental to a firm understanding of the method of models.

I think it is eminently helpful to point out to people not familiar with mathematical logic that proof is relative. I mentioned it myself about the Boolos piece. But then I realized I had read it too quickly and that it very much seems to me that he does intend provability to be relative to a theory (whatever theory one takes as encompassing the branches of mathematics) even if unspecified which theory. I allow him that especially since it is very common for people in foundations to say that ZFC axiomatizes the branches of mathematics. So I'm just saying, "Ward, don't you think you're being a little hard on the Beaver?"

that's not who the article is aimed at — fishfry

It seems it wasn't an article but something from a lecture. In any case, I would allow him some liberties when the purpose is to have some rhetorical fun rather than purporting to be an academically rigorous article.

"The whole of math" includes the ultimate truth or falsity of any given proposition, irrespective of its provability in any given axiomatic system. — fishfry

Yeah, I don't think that's what's mean by Boolos.

And it wouldn't make sense anyway to take him that way, since there is no known axiomatic theory that upholds such a collection of truths.

My opinion is supported by the fact that Kurt Gödel was a Platonist, and believed that there was a true fact of the matter for every mathematical proposition. — fishfry

That Godel was a platonist supports your opinion that Boolos's rhetorical lark is wrong and meaningless?

It is meaningful. People who understand that proof is relative, and who are already familiar with incompleteness, might appreciate that a brief spoken word bit doesn't have to provide all the qualifications.

what Boolos wrote was wrong. — fishfry

One source says that the bit comes from a lecture he gave. It might have been a lecture for logicians and students who would understand that he means whatever theory one might take as encompassing the branches of mathematics, whether it's ZFC or, for category theory, ZFC+inaccesible_cardinal, or whatever.

The piece is just a bit of fun, a bit of a stunt - outlining some deep mathematics with just one syllable words - obviously not pretending to include all the specifics.

Cantor's proof that there are more reals than natural numbers. The issue here is that the reductio ad absurdum proof [...] — ssu

Cantor's proof is not a reductio ad absurdum.

Cantor's proof can be outlined:

Show that there is no enumeration of the naturals onto the reals.

Show that any enumeration of the naturals is not onto the reals.

Let f be an enumeration of the naturals.

Blah blah blah and we've shown that f is not onto the reals.

So any enumeration of the naturals is not onto the reals.

So there is no enumeration of the naturals onto the reals.

my complaint is that [Boolos] did not distinguish between "PA can prove ..." and "It can be proved ..." — fishfry

Boolos says that he means proved by "the aid of the whole of math". My guess is that he means ZFC, which is ordinarily understood to provide an axiomatization for mathematics. So, as far as I can tell, he's talking about the second incompleteness theorem for ZFC.

(1) So in the empirical context, your objection was refuted.
— TonesInDeepFreeze

This is incorrect, because there is no empirical object referred to by "2+1", or "3". — Metaphysician Undercover

The question at that point was about Henry Fonda and names for Henry Fonda, not numbers and names for numbers. You objected to my Herny Fonda example onto itself.

You're lost in the conversation.

Now I'm waiting for proof that "2+1" refers to the same object as "3". — Metaphysician Undercover

You skipped again. I told you that proof of same denotation is finalized in the method of models. You don't have to wait around to find out about it - you can find it many books. As I said, as you also skipped:

"I can't cram it all into a post or even several posts. You need to read a book or other systematic presentation of mathematical logic in which the method of models is explained step by step, including the notions: concatenation functions, formal languages, signatures for formal languages, unique readability of terms and formulas, recursive definitions, mathematical induction, et. al. And prerequisite would be understanding basic mathematical notions, including: sets, tuples, relations, functions, et. al."

I wouldn't expect someone to teach me, in the confines of a posting forum, the subject of molecular biology. You would be foolish to think I should do it for you about mathematical logic. You could educate yourself!

. I do, nonetheless, intend to learn the language of formal logic — Aryamoy Mitra

I suggest this course of readings, in order:

Logic:Techniques Of Formal Reasoning - Kalish, Montague, and Mar

Elements Of Set Theory - Enderton (perhaps supplemented with Axiomatic Set Theory - Suppes)

A Mathematical Introduction To Logic - Enderton (definitely supplemented with the chapter on mathematical definitions in Inroduction To Logic - Suppes)

The Enderton logic book will take you through a proof of Godel-Rosser.

Godel's second incompleteness theorem explained in words of one syllable — bongo fury

We must keep in mind that when he says 'proved' it's really 'proved in [whatever particular theory we're talking about]'.

The insightful and witty George Boolos is one of the great writers about foundations - mathematical logic and set theory.

His 'The Logic Of Provability' might be the definitive treatment of the particular topics in that book.

He is a co-author of 'Computability And Logic', which is really nice classic overview.

He is a co-author of 'Logic, Logic, And Logic' which is a wonderful set of essays. Especially welcome is his cogent explanation of the intuitive basis of the set theory axioms in terms of hierarchy.

EDIT CORRECTION: Strike "We must keep in mind that when he says 'proved' it's really 'proved in [whatever particular theory we're talking about]'. He says that he means proved by "the aid the whole of math". I would take that to mean ZFC, which is ordinarily understood to provide an axiomatization for mathematics. So, as far as I can tell, he's talking about the second incompleteness theorem for ZFC.

I think the only translation of Godel's original paper approved by Godel is the one in Jean van Heijenoort's 'From Frege To Godel'.

Some people who have not very much background in mathematical logic have been able to understand incompleteness by first directly reading Godel's paper. But my native abilities would not allow me to understand the paper without having first studied incompleteness in textbooks in undergraduate level mathematical logic, and to get to mathematical logic I needed firs to study basic symbolic logic and set theory. Here are additional reasons I think it is better to learn from textbooks than from Godel's own paper.

* Godel-Rosser strengthens Godel's original and is what people actually mean now by incompleteness.

* Godel uses old-fashioned notation and terminology that is hardly used for many decades now. The newer notation, now decades established, is more streamlined and more aesthetically pleasing, and, for me at least, easier to understand. I think one is more conversant about incompleteness with the more modern notation and exposition.

In particular, since Godel's paper, the terminological distinction between recursion and primitive recursion became the prevailing convention, so some people get tripped up by Godel's older terminology.

The second incompleteness theorem is usually only sketched at the undergraduate level. And a full proof of the second incompleteness theorem is hard to find. It is in Hilbert-Bernays but it was only recently translated from German to English. But there are still some good undergraduate level books about the second theorem. Also, Godel didn't provide much proof of the Representation theorem that is a crucial lemma for incompleteness. I've read that a proof of the Representation theorem is a lot of tedious technical detail. If I'm not mistaken, it also is in Hilbert-Bernays.

The aforementioned Peter Smith book is really good, but again, I think it is better appreciated by first studying basic symbolic logic, set theory, and mathematical logic.

And for an everyman's introduction, Torkel Franzen's book is superb and a joy to read. Though again, personally, I doubt I would have appreciated it as much as I did without having first studied mathematical logic.