Then why do you ask me to repeat myself? — Metaphysician Undercover

He didn't.

To begin with in all that, what's your definition of "real thing"?

I don't like that tree. The order and proximities seem incorrect and haphazard in places.The Venn diagram is better.

As

I've been on grants before and even at the time felt it was not a productive use of public monies. — jgill

As a taxpayer, I'm quite happy for my share - my .00002 cents - to have contributed to your research. More, if I had it to give.

There's an interesting angle on this.

In the language of PA we can express.

Sentence S can't be negated.

It's false, but it can be stated in the language.

And, I'm not sure, but I suspect we can have:

The sentence with Godel-number n can't be negated.

And also have the above sentence have Godel-number n.

And "the sentence with Godel-number n can't be negated" would be false, as seen by the fact that, contrary to what the sentence claims, we would simply show that the negation of "the sentence with Godel-number n can't be negated" is also a sentence in the language of PA.

But if a contradiction in PA could be derived from this, then that would prove the inconsistency of PA.

Some brilliant mathematicians have spent a large part of their lives trying to prove (contrary to mathematical consensus) that PA is inconsistent.

If anyone proves that PA is inconsistent then it would be huge headline news, not just in mathematics but generally. It would be "earth shaking". Now, that is not itself an argument that the TheMadFool's exercise is not correct, but it puts this in perspective that one would be extremely doubtful that his argument to his conclusion could be made correct even with needed redaction.

Bottom line for your exercise:

'This sentence can be negated' is true and not paradoxical.

'This sentence can't be negated' is false and not paradoxical.

My guess is it is not easy, even if possible, to get a paradox from merely syntactical considerations ('sentence', 'negation', et. al). Paradoxes usually arise from semantical considerations ('true', 'false', 'definable', et. al).

Your exercise is not in a mathematical context, which is okay, but it's worth noting comparison with mathematics (I'm simplifying here).


Consider:

This sentence is not provable.

There is only one 'this sentence' in Godel's argument.

The "self reference" of 'this sentence' is okay, because the actual formal sentence doesn't use 'this sentence'. It is paraphrased more fully:

The sentence with Godel-number n is not provable in theory T.

And the above sentence has Godel-number n.


Consider:

This sentence is false.

There is only one 'this sentence' in Tarski's argument.

But the actual formal sentence doesn't use 'this sentence'. It is paraphrased more fully:

The sentence with Godel-number n is false in a model of theory T.

And the above sentence has Godel-number n.

And Tarski proves that if 'is false in a model of theory T' can be defined within the theory T, then the theory is inconsistent. So it's not even a matter whether the "self-reference" of 'this sentence is false' is okay; rather, in a consistent theory, we are not even capable of saying 'this sentence is false'.

"IF this (1) statement can be negated THEN this (2) statement can't be negated" — TheMadFool

Putting '(1)' between 'this' and 'statement' is not coherent. And putting '(2)' between 'this' and 'statement' is not coherent.

Maybe you mean:

If "this statement can be negated" is true, then "this statement can't be negated" is true.

Or with the 'N' you used in a previous edit:

'N' stands for 'this statement can be negated'.

Then your premise 3. is:

If N then ~N.

And, since N is also 'N can be negated', 'If N then ~N' is 'If N can be negated then N cannot be negated'.

And then there is no ambiguity with 'this statement'.

Or, using your '(1)' and '(2)':

'(1)' stands for 'this statement can be negated'.

'(2)' stands for 'this statement can't be negated',

Then your premise 3. is:

If (1) then (2).

And 'this statement' in (1) denotes 'this statement can be negated'. And 'this statement' in (2) denotes 'this statement can't be negated'.

And then there is ambiguity with 'this statement'.

P.S.

In Argument A, 2. does not need to be taken as a premise. 2. follows from 1. by UI.

If you wish to revise further, then please state any further amended arguments in new posts, so that my replies are still pertinent relative to the posts I replied to.

Argument A

1. All statements can be negated [assume for reductio ad absurdum]

2. If all statements can be negated then this statement can be negated [premise]

3. If this statement can be negated then this statement can't be negated [premise]

4. If this statement can't be negated then not all statements can be negated [premiseMP]

5. This statement can be negated [1, 2 MP]

6. This statement can't be negated [3, 5 MP]

7. Not all statements can be negated [4, 7 MP]

8. All statements can be negated AND not all statements can be negated [1, 2 Conj]

9. Not all statements can be negated [1 - 8 reductio ad absurdum]

QED

Now, consider the statement, This statement can't be negated

Argument B

1. Either this statement can't be negated can be negated or this statement can't be negated can't be negated [premise]

2. If this statement can't be negated can't be negated then this statement can't be negated can't be negated [premise]

3.. If this statement can't be negated can be negated then this statement can be negated [premise]

4. If this statement can be negated then this statement can't be negated [premise]

5. This statement can't be negated can be negated [assume for conditional proof]

6. This statement can be negated [3, 5 MP]

7. This statement can't be negated [4, 6 MP]

8. If this statement can't be negated can be negated then this statement can't be negated [5 - 7 conditional proof]

9. This statement can't be negated can't be negated or this statement can't be negated [1, 3, 8 CD]

QED — TheMadFool

I'll mention the first problem I find in each Argument, before going on to the rest of it:

Argument A:

3. If this statement can be negated then this statement can't be negated [premise] — TheMadFool

That is a false premise.

Also, "this statement" in that premise denotes "If this statement can be negated then this statement can't be negated", but "this statement" in line 2 denotes "If all statements can be negated then this statement can be negated". So "this statement" is used ambiguously in the argument.

And if "this statement" weren't ambiguous, then 3. is just the negation of 2, while 2. comes from 1. So 1. and 3. are inconsistent. So, of course, we can derive a contradiction is we assume both 1. and 3. There would be no point in your excercise.

Argument B:

1. Either this statement can't be negated can be negated or this statement can't be negated can't be negated [premise] — TheMadFool

"this statement can't be negated can be negated" is not grammatical, so I don't know that it is supposed to mean.

Maybe you mean:

"this statement can't be negated" can be negated.

And that is true. And "this statement can't be negated" is false.

"this statement can't be negated can't be negated" is not grammatical.

Maybe you mean:

"this statement can't be negated" can't be negated.

And that is false. And ""this statement can't be negated" can be negated" is true.

That's a major re-edit of the OP after several edits. Before I reply, is that your final edit?

Every sentence can be negated, simply by putting a negation sign in front of the sentence. Doing that is purely a syntactical operation. It does not mean that we are asserting the negation.

So "This sentence can be negated" is true. That is, N is true since N can be negated by writing

~N

/

N = This sentence can be negated.

~N = This sentence can't be negated — TheMadFool

There's slippage there in what 'this sentence' refers to.

In "This sentence can be negated", "this sentence" refers to "This sentence can be negated".

But in "This sentence can't be negated", "this sentence" refers to "This sentence can't be negated".

So "this sentence" refers to two different things in your writeup.

That is just a foible of English that "this" changes meaning by context.

So, to avoid ambiguousness, you probably have to reformulate "This sentence can't be negated" without "this sentence".

Then we can see how the rest of your argument fares.

I wouldn't be surprised at all if there were some striped kangaroos. — Down The Rabbit Hole

I think you mean you would be surprised.

I just don't think we should lower our burden of proof based upon the difficulty of obtaining evidence. — Down The Rabbit Hole

That seems reasonable.

On the other hand, if an outlandish or "out of thin air" existence claim is asserted, it doesn't seem reasonable that the denier would have as great a burden to prove false as the assertor has to prove true.

/

Or consider recent history. A lot of Republicans, including the president, claim there was widespread voting fraud, but they have not produced convincing evidence. People who recognize the legitimacy of the election don't just say that the claim has no evidence but moreover that the claim is to be regarded as false.

This would be using evidence to reach a conclusion. — Down The Rabbit Hole

You mentioned "sufficient evidence". I'm wondering whether we would deem my knowledge of kangaroos to be sufficient to have a reasonable belief that there do not exist striped kangaroos. If not then you would place a burden of proof on me if I say, "Striped kangaroos do not exist". If you did, then you might be right; I don't know.

The more poignant case is something like "There exists a billion ton being with intelligence, knowledge, physical strength, and moral purity a billion times greater than any human being."

If I say, "I don't believe that", then we agree that is reasonable.

But if I say, "That is false", then I have a burden to prove it is false?

"There exists a fish with blue fins and a green body."

I don't assume that is true and I don't assume that it is false.

"There exists a striped kangaroo."

I assume that is false.

Interesting that Algebra dominates. I would have thought it would be Analysis and its offshoots.

you need a reason to believe something. If there is no reason, then disbelief is warranted. That is to say that the truth of the belief in question can be rejected, or denied.
— Pinprick

To actively claim something does not exist, you have a burden of proof — Down The Rabbit Hole

There is a difference between "S is false" and "I disbelieve S."

"S is false" in many contexts raises expectation of demonstration that S is false.

But "I disbelieve S" is merely an assertion that I am either withholding or rejecting belief that S is true. If one has not been shown adequate demonstration that S is true, then it may be reasonable to withhold or reject belief that S is true.

If someone says "There are kangaroos living at the North Pole", then I may say "I don't believe there are kangaroos living at the North Pole" without obligation of proving that there are no kangaroos living at the North Pole.

One more time:

We are in a context of everyday English. Then, I have given a mathematical representation of that everyday English sense. A mathematical representation:

A count is a bijection f from a finite set onto a set of successive positive numbers that includes 1. The result of the count is the greatest number in the range of the bijection. The count induces an order on the domain by: x precedes y iff f(x) < f(y).

{<'War And Peace' 1> <'Portnoy's Complaint' 2>} is a count.

The domain is {'War And Peace' 'Portnoy's Complaint'}.

The range is {1 2}.

The result is 2.

The order induced is {<'War And Peace' 'Portnoy's Complaint'>}.

Why do you keep avoiding the question? — Metaphysician Undercover

Why do you keep saying I've avoided the question when I have not, when, indeed, I have answered several times and with copious explanation and detail? Possible answers: (1) You are dishonest, (2) You have cognitive problems.

within a logical system you cannot change the "sense" of a word without the fallacy of equivocation — Metaphysician Undercover

In a formal system, a terminology can be defined only once, so it is not possible to have equivocation.

If we are going to say that zero objects is a countable number of objects, then we need a definition of "count" which is consistent with this. — Metaphysician Undercover

I didn't use the phrase "countable number of objects".

None. We know that there are zero, without counting any. — Metaphysician Undercover

Correct. I told you that the pedantic technical mention I made does not pertain to the everyday English sense of 'count'.

if we can assign such a value to imaginary things in a similar way, we need a principle to establish equality, or compatibility, between observed things and imaginary things. This is required to use negative numbers. — Metaphysician Undercover

That is an extraordinary statement, even for you.

Possible outcomes:

ExBx

Black dog found before end. $500 and get to go home early.
Black dog found at end. $500.
Black dog not found. Wasted time trying.

~ExBx

Black dog found before end. Wasted time trying, but get to go home early.
Black dog found at end. Wasted time trying.
Black dog not found. $500.

If black dog found before end, advantage goes to ExBx.
If black dog found at end, advantage goes to ExBx.
If black dog not found, advantage goes to ~ExBx.

So advantage overall goes to ExBx, because it might be that ExBx has a chance to win $500 and go home early, while ~ExBx has only a chance to win $500 after going the full distance.

/

Put another way,

Suppose the chance of ExBx is the same as the chance of ~ExBx (i.e. we don't have reason to believe in advance that one is more likely than the other).

The expectation of the work to prove ExBx is less than the expectation of the work to prove ~ExBx, since proving ExBx might finish earlier than the end of the search, while proving ~ExBx will surely not finish earlier than the end of the search.

I think there are infinite things we can prove with math and infinite things we cant. — Gregory

For any given consistent system S, there are infinitely many theorems and infinitely many non-theorems.

I'd be curious what the percentages are for each of the branches. To not have it too Balkanized, it would have to be large groupings like:

Foundations, Algebra, Analysis, Topology, Geometry, Number Theory, Combinatorics. Graph Theory, Probability and Statistics, Applied, Game Theory

That chart seems to capture discovery not proof. For example, the min in row 4 is 1 only because we discover that there is a black dog and give up trying to prove that there is not one. But that is not the task. The task is to prove there is not a black dog.

Suppose someone says to you:

"I have two stacks of photographs. Each stack has 10000 pictures of dogs. In one stack there's at least one picture of a black dog. In the other stack there is no picture of a black dog. I am going to randomly give you one of the stacks. Now you have a choice:

You can choose to prove there is a picture of a black dog by going through the pictures until you find a picture of a black dog and then you may stop, and I pay you $500. But if you don't find a picture of a black dog to prove that there is one, then I pay you nothing.

or

You can choose to prove there is no picture of a black dog by going through all the pictures and not finding a picture of a black dog, and I pay you $500. But if you do find a picture of a black dog, then you may stop, but I pay you nothing."



The chance of being paid is even between the two choices. But, clearly, one should choose the best chance at having the shortest labor time - by choosing to prove there is a picture of a black dog. Because, if there is a picture of a black dog, then you get to quit when you find it. But if you choose to prove there is no picture of a black dog, and you don't find one, then you don't get to quit until you've gone through all the pictures.

In other words: If you choose "there is a picture of a black dog" then you can both win and have a chance of quitting early, even as early as looking at the first picture. But if you choose "there is no picture of a black dog" then you can't win unless you look through all the pictures.

If someone told me they chose "prove there is no picture of a black dog", I'd say "You must really like going through pictures of dogs, or you don't value your time, or you're stupid."

/

And I realize that this works no matter whether the domain is determinate or indeterminate (it's just that it's even worse for ~Ex Bx when the domain is indeterminate).

https://en.wikipedia.org/wiki/Proof_by_contradiction#Irrationality_of_the_square_root_of_2 — InPitzotl

Here are some more problems with that article:

(1) The sqrt(2) proof does not make clear its indirect form. Indirect in clear form would be to assume "~ sqrt(2) is irrational". But the proof assumes "sqrt(2) is rational".

Basically the same objection is mentioned by a participant in the Talk section for the article.

(2) Intuitionistic invalidity is mentioned only in passing as a clause in a sentence about excluded middle. The very important distinction, vis-a-vis intuitionism, between the two forms of deriving a contradiction is not mentioned. (That is, the article does not mention that "Suppose P, derive contradiction, infer ~P" is intuitionistically valid.)

And the article says "some intuitionist mathematicians do not accept [excluded middle]". Why only say 'some', why not say 'all'? (Or maybe the author knows of intuitionists who accept excluded middle?)

(3) The article mentions Cantor's diagonal argument as proof by contradiction. I don't recall whether Cantor himself used indirect proof (I tend to think he didn't), but even if he did, it should be noted that his argument does not require indirect proof and it is intuitionistically valid.

(4) There are a few good citations in the list of references, but some of them mention proof by contradiction only tangentially. And the rest of the list is lousy, including quite informal lecture bullet points and things like that.

/

What is the importance in math forums for keeping the distinction between the two forms clear? It is that often cranks disparage proofs by such as Cantor on the basis that they're "indirect" while these cranks have heard somewhere that indirect proofs are suspect (indeed, they are worse than suspect for intuitionists). But the proofs are actually not indirect, or if they are, they can be rearranged so that they are not indirect.

If you count "1", then it is implied that there is one thing (an object) counted. Do you, or do you not agree with this?
— Metaphysician Undercover

Agree. — TonesInDeepFreeze

That was a while ago. But you're still asking!

I think you agree with me on the necessity of having two objects to make the use of "2" or "second", a true or valid use. — Metaphysician Undercover

There you even correctly posted yourself that you surmise that I agree that if we count 2 objects then there exist 2 objects.

The context here has been of a shelf that has books on it. I've said more than once that if you count 1 book or 2 books, then, yes, there are books on the shelf.

From this thread, this is the context in which we are talking about a shelf that has books on it:

To have a count of one, there must be an object which is counted. In order for the count to be a valid count, there must be something which is counted. — Metaphysician Undercover

Do you agree that there must be some of these things (objects) which are classed as "books", for us to have a true count.
— Metaphysician Undercover

I've answered that already a few times. To have a non-empty count, of course there exist the objects counted, and in you example, these objects are books. — TonesInDeepFreeze

Do you agree that there is no activity of counting if there is no objects counted?
— Metaphysician Undercover

Now I'm answering yet again, there is no no-empty count if there are not objects counted. — TonesInDeepFreeze

Therefore the number 5 loses its meaning if it does not refer to five of something counted, books in this case. — Metaphysician Undercover

The context is not a shelf with no books, but a shelf with 5 books.

In everyday understanding, when we count, we associate one thing with 1, then the next thing with 2, etc. — TonesInDeepFreeze

There I keep in context of having at least one book on the shelf.

To have a count of one, there must be an object which is counted. In order for the count to be a valid count, there must be something which is counted. — Metaphysician Undercover

Again, the context is that there are books on the shelf.

The count of two is justified by the existence of two such objects — Metaphysician Undercover

Again, the context is that there are books on the shelf.

Now:

If no books are counted, do you consider this to be a count? — Metaphysician Undercover

I already addressed that. If there are no books, then it's not a non-empty count. It's not the kind of count we're talking about in your example.

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.) — TonesInDeepFreeze

And that parenthetical is simply to make clear that in this context we're not talking about the technical notion of an empty count. We're talking about counts that start at 1.

/

I'll say it one more time in this forrm: If there is a count that reaches 1, then there exists at least one object counted, and if there is a count that reaches 2, then there exist at least two objects counted.

And that reflects the representation with a bijection. If a natural number n is in the range, then there must be at least n objects in the domain.

You don't read my posts adequately to register in your mind what I wrote, let alone understanding them.

And you're even more ridiculous, since the question of whether there are objects counted - already answered by me - is answered right in the representation with a bijection itself. You can see for yourself that the two books are right there in the domain of the bijection.

/

I'm asking you if you believe there is such a thing as an empty count — Metaphysician Undercover

Your original and ongoing question regarded the context in which there are books on the shelf. You didn't ask me about the notion of an empty count.

I mentioned the empty count only to avoid a pedantic, technical hitch. I am not talking about empty counts in the context where there are books on the shelf.

But about the empty count: It's a technical set theoretical matter. It's not intended that the use of the word 'count' in 'empty count' corresponds to our everyday English senses of 'count'. I happily agree that it's an odd use of the word 'count'. If you don't like the notion, then that's okay in this context, because the representation with a bijection doesn't depend on the notion.

/

you are counting hypothetical doors, symbolic representations of doors — Metaphysician Undercover

But it's still counting.

If you present this as a true count of actual captains of an actual starship, you'd be engaged in deception. — Metaphysician Undercover

Ah, you resort to the strawman. We are not claiming it is a count of actual captains.

fractions for dividing up a pumpkin pie — fishfry

Not in the real world. I eat the whole pie all at once.

Math needs to correspond to reality!

Definitions:

x is rational iff x equals a ratio of integers

x is irrational iff ~ x is rational

Theorems:

x is irrational iff ~ x equals a ratio of integers

if ~ x is irrational then x is rational (not intuitionistically acceptable)


(1) It's indirect if you put it this way:

Prove sqrt(2) is irrational

Suppose ~ sqrt(2) is irrational

Derive contradiction

So sqrt(2) is irrational (not inuitionistically acceptable)


(2) But that's not necessary, and it can be done without indirect proof (intuitionistically acceptable) this way:

Prove sqrt(2) is irrational

Prove ~ sqrt(2) is rational

Suppose sqrt(2) is rational

Derive contradiction.

So ~sqrt(2) is rational

So sqrt(2) is irrational

I took out what I previously posted in this message. My questions were not needed.

relying on the assumption that said pattern repeats indefinitely by virtue of being a pattern. — Benj96

Proof that certain reals have infinite expansions does not rely on assumptions that patterns repeat.

so the only other way to determine it in the absolute is the exhaustive method - which is an endless endeavour in said case. — Benj96

No, it is not endless. Proofs in mathematics are what you call "exhaustive" in finite length.

There’s no proofs we can do to determine if indeed it continues as 3 forever — Benj96

No, it is proven.

We have no base lines - it’s all arbitrary. — Benj96

Perhaps you might say that the axioms are arbitrary. But, given the axioms, proof that some reals have infinite decimal expansions is definite.

I think you're focused too much on proof by contradiction.
— InPitzotl

[it's] the only method which makes proving a negative easier than proving the positive. — TheMadFool

As mentioned, 'proof by contradiction' is not the right term. And such cases can more comprehensively be described as 'deductively proving'. We may prove ~ExP deductively, either by definition (proving that there are no married bachelors), or from axioms or principles (proving that there is no rational number whose square is 2), or from facts taken as premises (proving that there is not a horse in the refrigerator from the premise that horses are not smaller than the space inside the refrigerator).

For deductive proofs of ~ExP, usually the method is to assume ExP, then derive a contradiction, then conclude ~ExP. This usually deploys modus tollens in this form: ((ExP -> Q) & ~Q) -> ~ExP, which is permitted by direct proof.

So I would restate your claim as: Other than deduction, there are no methods that make it easier to prove ~ExP than to prove ExP. (I will leave it as tacit that in such comparisons that we are concerned with relative difficulty only in context of which is true. If ExP is false, then necessarily it's easier to prove ~ExP, and if ~ExP is false, then necessarily it's easier to prove ExP.)

But "[it's] the only method which makes proving a negative easier than proving the positive" is not self-evidently true. It requires an argument. It is an ~ExP claim ("there does not exist a method that is neither deduction nor case-by-case in an indeterminate domain"), so notice that - contrary to your claim that ExP is necessarily claimed before ~ExP - this is an example where the ExP claim was not made first.

In regard to difficulty in re existential claims that pertain to the physical, it goes without saying they're much easier to prove than their negations but, as your example shows, positive existential claims that are amenable deduction are sometimes harder to demonstrate than their negations. — TheMadFool

The deductions can be about physical facts. From premises about physical facts we may deductively reach conclusions.

proving the positive, particular affirmative (Some A are B) is definitely easier than proving the negative, universal negation (No A are B). Experts agree on that and I defer to their expertise. — TheMadFool

(1) I will regard that in the context "except for deductions), (2) Your claim depends on whether there are methods other than deduction and case-by-case in an indefinite domain.

Insofar as categorical statements are the issue, proving the positive, particular affirmative (Some A are B) is definitely easier than proving the negative, universal negation (No A are B). Experts agree on that — TheMadFool

Unless you tell us the arguments of these experts, it's mere appeal to authority. So who are these experts and where can I read their arguments?

the difficulty of a proof is proportional to the number of cases you have to test by that method. The weakness of this approach is simply that it only applies when you're using "proof by testing each case". — InPitzotl

Right, case-by-case in an indeterminate domain.

The irrationality of [the square root of] two can be demonstrated using proof by contradiction — InPitzotl

The ordinary proof that the square root of 2 is irrational is not a proof by contradiction. Assuming P, deriving a contradiction, then inferring ~P is not a proof by contradiction. It might seem that deriving a contradiction on the way to the conclusion is proof by contradiction, but 'proof by contradiction' refers to something more specific: Assuming ~P, deriving a contradiction, then inferring P.

If I'm trying to show there aren't any black dogs, but it turns out there are, I still stop early once I find the black dog. — InPitzotl

But that is not an instance of demonstrating that there are no black dogs. It is better described as the process of discovery whether there are black dogs. That's different from demonstrating that there are no black dogs.

Proof by contradiction/indirect proof — TheMadFool

Be careful with the terms 'proof by contradiction' and 'indirect proof'.

This is the form of proof by contradiction (indirect proof):

Assume ~P
Derive contradiction
Infer P

This is not a from of proof by contradiction (indirect proof):

Assume P
Derive contradiction
Infer ~P

since positive claims precede their negation (~p can be only after p) and since to assert a proposition one needs proof, it follows that positive claims need to be proven first. — TheMadFool

No, it does not follow. I've given you explanations for why it does not follow. You skip responding to the key points in the explanations.

Do you agree that there must be some of these things (objects) which are classed as "books", for us to have a true count. — Metaphysician Undercover

I've answered that already a few times. To have a non-empty count, of course there exist the objects counted, and in you example, these objects are books.

Do you agree that there is no activity of counting if there is no objects counted? — Metaphysician Undercover

Now I'm answering yet again, there is no no-empty count if there are not objects counted.

Now, are you going to continue asking me this over and over again?

find some agreement or compromise — Metaphysician Undercover

I don't seek agreement or compromise. I'm interested in showing where your remarks are incorrect, especially ignorant and/or confused, and sometimes also to add explanations about mathematics, whether you ever understand them.

when I said "moving in infinite circles" I meant always trying to prove things in a system where you don't know if what your trying to prove can even be proven — Gregory

For a consistent formal (I always mean 'consistent formal' in this context) system S, there is an infinite enumeration of the proofs. So it is linear, not circular.


In a idealized context without a finite upper limit of time to prove (such as with Turing machines), if S is incomplete, then for a sentence P, at any point in the enumeration, there are these possibilities (proof relative to S):

(1) P been proven.

(2) ~P has been proven.

(3) P is provable, and we will eventually find a proof.

(4) ~P is provable and we will eventually find a proof.

(5) P is not provable and ~P is not provable, and we will never find a proof of P and we will never find a proof of ~P ("eternally floundering to know" whether P is provable and "eternally floundering to know" whether ~P is provable).

(6) In a metatheory for S, we prove "P is provable or ~P is provable".

(7) In a metatheory for S, we prove that P is provable though we don't know a proof itself.

(8) In a metatheory for S, we prove that ~P is provable though we don't know a proof itself.

(9) In a metatheory for S, we prove that P is not provable. (e.g., Cohen proof that AC is not provable from ZF and that CH is not provable from ZFC)

(10) In a metatheory for S, we prove that ~P is not provable. (e.g., Godel proof that ~AC is not provable from ZF and that ~CH is not provable from ZFC)

(11) In a metatheory for S, we prove that P is not provable and that ~P is not provable. (e.g., the conjuction of Godel and Cohen)


For mortal beings, or assuming that in some finite time there won't' be any conscious beings, "eventually" does not hold.

But mathematics itself is very clear about the limitations mentioned in (1)-(11) and makes that lack of conclusiveness itself a subject of rigorous study. This is yet another respect in which mathematics is diametrically different from religion.

All mathematicians says that numbers exist in some sense, which is the same as saying they have truth value. — Gregory

No, what have truth values are statements.

What have a truth values are the statements "There exists a natural number that is the successor of 0" and "There exists a natural number that is the successor of 1".

The numbers themselves don't have truth values.

This is not a mere pedantic distinction. It is a distinction needed so that the discussion about this subject is coherent.

Searching for logical certainty — Gregory

The theorems are certain or uncertain exactly to the extent that the axioms are certain or uncertain.

It is freely granted that the common axioms are non-logical. That is why it is really stupid for you to say that I'm a logicist (or however exactly you said it).

However, if any non-logical mathematical judgements are certain, then they are those of finitistic combinatorial mathematics. We may be skeptical of finitistic combinatorial mathematics, but then we might as well be skeptical of everything mathematical.

1 plus 1 equals two only if 1 and 2 exist and can exist. So things boil down to our world view at the end of the day — Gregory

I asked whoever said that numbers have truth value(s)?

Or is it your own claim that numbers have truth value(s)?

How do you know numbers have truth value? — Gregory

Whoever said numbers have truth value(s)?

Those two sentences together comprehensively state Gödel system without using equations — Gregory

The second one is reasonable. The first one is a mess.

I think you are being too literal. — jgill

Too literal about the word 'religious'? It is used as cudgel. So it is good, as a starting point at least, to point out that it is not literally true.