There are no "true statements' in mathematics

[ladyjane]

faust says

Janie - what I am saying is that there are no unqualifiedly true statements

so then godels incompleteness theorem is just meaningless babble-as colin leslie dean has always said- as he cant tell us what he means by true statement in his theorem

Gödel’s first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

[Faust]

Every mathematical truth rests upon assumptions. Godel was supposed to have found a rejoinder to Russell, but Russell was quite aware that Peano’s axioms were not provable, which is why they are axioms. Rather, he confirmed Russell’s suspicions. That was an advance, of a sort, I guess. There is nothing inconsistent between Godel and your friend Dean in this regard.

Peano’s axiom that equality is reflexive is what Russell called “self-evident” - but this is not a proof of anything. Similarly with the “symmetry” of equality. Peano’s innovation was not to prove anything, but to show that these (all of them) axioms were actually necessary to mathematics.

In that sense, Godel added nothing. Russell, for instance, already knew that the defintion of “number” itself relied on a certain self-evidence. But so does the defintion of everything else.

[ladyjane]

faust says

Every mathematical truth rests upon assumptions

but also says

Janie - what I am saying is that there are no unqualifiedly true statements

which confirms colin leslie deans argument that godels incompleteness theorem is meaningless babble as he cant tell us what true statements are

Gödel’s first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

faust says

Godel was supposed to have found a rejoinder to Russell, but Russell was quite aware that Peano’s axioms were not provable,

the point is godel made a distinction between provable and true statement
the question is not whether peano is provable but weather he is true
and godel does not tell why or how peano is true -or any other mathematical statement

[ladyjane]

faust says

Every mathematical truth rests upon assumptions

but also says

Janie - what I am saying is that there are no unqualifiedly true statements

which confirms colin leslie deans argument that godels incompleteness theorem is meaningless babble as he cant tell us what true statements are

Gödel’s first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

faust says

Godel was supposed to have found a rejoinder to Russell, but Russell was quite aware that Peano’s axioms were not provable,

the point is godel made a distinction between provable and true statement
the question is not whether peano is provable but weather he is true
and godel does not tell why or how peano is true -or any other mathematical statement

[denali]

Applying the notion of “truth” to it is already an unnecessary interpretation of an entirely rigorous proof. We should not be so seduced by natural language.

[ladyjane]

Faust says

Godel was supposed to have found a rejoinder to Russell, but Russell was quite aware that Peano’s axioms were not provable,

but ponicare and russell argued peano axioms where invalid as they are impredicative and impredicative statements are according to text books on logic invalid
so godel is again invalid as he uses invalid statements as shown by colin leslie dean

gamahucherpress.yellowgum.com/bo … GODEL5.pdf

”This sense of definition allowed Poincaré to argue with Bertrand Russell over Giuseppe Peano’s axiomatic theory of natural numbers.

Peano’s fifth axiom states:

• Allow that; zero has a property P;
• And; if every natural number less than a number x has the property P then x also has the property P.
• Therefore; every natural number has the property P.

This is the principle of complete induction, it establishes the property of induction as necessary to the system. Since Peano’s axiom is as infinite as the natural numbers, it is difficult to prove that the property of P does belong to any x and also x+1. What one can do is say that, if after some number n of trails that show a property P conserved in x and x+1, then we may infer that it will still hold to be true after n+1 trails. But this is itself induction. And hence the argument is a vicious circle.

From this Poincaré argues that if we fail to establish the consistency of Peano’s axioms for natural numbers without falling into circularity, then the principle of complete induction is improvable by general logic.

[Faust]

Russell did not argue that the axioms are invalid. I don’t know where you get that from. “Invalid” can’t be exactly the term you mean - statements are never invalid (or valid), but Russell didn’t argue against them, in any event.

It is not “difficult to prove” the fifth axiom - it is impossible to definitively show that it is true.

There is a difference.

And yes, induction is not provable by deduction.

But again, we have known this for years.

And years.

Have fun.

[ladyjane]

It is not “difficult to prove” the fifth axiom - it is impossible to definitively show that it is true.

There is a difference.

And yes, induction is not provable by deduction.

But again, we have known this for years.

And years.

fact is godel used an impredicative statements pointed out by ponicare and russell and such statements are invalid according to text books on logic
peano axioms are invalid as they are circular and circular statements are considered invalid in text books on logic

from wiki
en.wikipedia.org/wiki/Begging_the_question

In logic, begging the question has traditionally described a type of logical fallacy (also called petitio principii) in which the proposition to be proved is assumed implicitly or explicitly in one of the premises.[1][2][3][4][5][6] Begging the question is related to the fallacy known as circular argument, circulus in probando, vicious circle or circular reasoning. The first known definition in the West is by the Greek philosopher Aristotle around 350 B.C., in his book Prior Analytics.

http://en.wikipedia.org/wiki/Preintuitionism

”This sense of definition allowed Poincaré to argue with Bertrand Russell over Giuseppe Peano’s axiomatic theory of natural numbers.

Peano’s fifth axiom states:

• Allow that; zero has a property P;
• And; if every natural number less than a number x has the property P then x also has the property P.
• Therefore; every natural number has the property P.

This is the principle of complete induction, it establishes the property of induction as necessary to the system. Since Peano’s axiom is as infinite as the natural numbers, it is difficult to prove that the property of P does belong to any x and also x+1. What one can do is say that, if after some number n of trails that show a property P conserved in x and x+1, then we may infer that it will still hold to be true after n+1 trails. But this is itself induction. And hence the argument is a vicious circle.

From this Poincaré argues that if we fail to establish the consistency of Peano’s axioms for natural numbers without falling into circularity, then the principle of complete induction is improvable by general logic. “

[ladyjane]

FAUST PLEASE RESPOND

faust says

Every mathematical truth rests upon assumptions

but also says

Janie - what I am saying is that there are no unqualifiedly true statements

which confirms colin leslie deans argument that godels incompleteness theorem is meaningless babble as he cant tell us what true statements are

Gödel’s first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

[Faust]

Yes, ladyjane - yeeeeeeeeessssssssss!

Induction is unprovable.

We knoooooooooooooowwwwwwwwwwww.

Axioms themselves are not circular. Arguments are. You should really get to know the vocabulary if you’re going to talk about this stuff.

Axioms are taken as self-evident, as i have said. But not circular. They are taken as unprovable.

I think you have a basic misunderstanding of what an axiom is. Maybe Godel did, too, but it’s okay - it may have led to something, anyway.

An axiom is a minimal assumption. If it weren’t an assumption, it would cease to be an axiom. It’s as simple as that. If it were provable, it wouldn’t be an assumption or an axiom. If it weren’t self-evident, it wouldn’t be an axiom.

Maybe while you’re wiki-ing stuff, you should just wiki “axiom”.

This is all a tempest in a teapot.

[ladyjane]

Axioms themselves are not circular. Arguments are

sorry peanos 5th axiom is circular

Peano’s fifth axiom states:

• Allow that; zero has a property P;
• And; if every natural number less than a number x has the property P then x also has the property P.
• Therefore; every natural number has the property P.

This is the principle of complete induction, it establishes the property of induction as necessary to the system. Since Peano’s axiom is as infinite as the natural numbers, it is difficult to prove that the property of P does belong to any x and also x+1. What one can do is say that, if after some number n of trails that show a property P conserved in x and x+1, then we may infer that it will still hold to be true after n+1 trails. But this is itself induction. And hence the argument is a vicious circle.

Axioms are taken as self-evident, as i have said. But not circular. They are taken as unprovable.

sorry
euclids 5th was shown to be wrong
the axiom of rediciblity is invalid according to most mathematicians
and godel used it in his proof which makes his proof invalid

[Faust]

Dude - that’s not actually Peano’s axiom.

This is boring.

[ladyjane]

Dude - that’s not actually Peano’s axiom.

bookrags.com/research/peano-axioms-wom/

Axiom 5. If S is a set of numbers containing 0 and if the successor of any number in S is also in S, then S contains all the numbers.

bookrags.com/research/peano-axioms-wom/

However, Peano’s fifth axiom, the principle of mathematical induction gives us the means for determining whether property P is true for all numbers in {0, 1, 2, 3…}. The induction axiom says that we need carry out only two steps to determine the truth of P for the infinite set of whole numbers. First, we must show that 0 has property P. Second, we must show that, for any number n, if n has property P, then this implies that the succesor of n, namely n+1, also has property P.

en.wikipedia.org/wiki/Preintuitionism

Peano’s fifth axiom states:

• Allow that; zero has a property P;
• And; if every natural number less than a number x has the property P then x also has the property P.
• Therefore; every natural number has the property P.

HOW ABOUT RESPONDING TO THIS
FAUST

faust says

Every mathematical truth rests upon assumptions

but also says

Janie - what I am saying is that there are no unqualifiedly true statements

which confirms colin leslie deans argument that godels incompleteness theorem is meaningless babble as he cant tell us what true statements are

Gödel’s first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves basic[b] arithmetical truths[/b], an [b]arithmetical statement that is true[/b], but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

[denali]

sorry
euclids 5th was shown to be wrong

No, it was shown that there are useful geometries which do not use it.

which confirms colin leslie deans argument that godels incompleteness theorem is meaningless babble as he cant tell us what true statements are

The proof itself does not need “truth” at all: it stands on its own. Please learn to distinguish proofs from their interpretations.

the axiom of rediciblity is invalid according to most mathematicians

Do cite.

Peano’s fifth axiom states:

• Allow that; zero has a property P;
• And; if every natural number less than a number x has the property P then x also has the property P.
• Therefore; every natural number has the property P.

This is the principle of complete induction, it establishes the property of induction as necessary to the system. Since Peano’s axiom is as infinite as the natural numbers, it is difficult to prove that the property of P does belong to any x and also x+1. What one can do is say that, if after some number n of trails that show a property P conserved in x and x+1, then we may infer that it will still hold to be true after n+1 trails. But this is itself induction. And hence the argument is a vicious circle.

I’m really not sure what that interpretation is getting at. There’s no actual induction going on in a proof by mathematical induction.
If you don’t understand the formal meaning of Peano’s Axioms, please refrain from thinking you can deconstruct mathematics based on them. Axioms aren’t “arguments”; “circularity” is impossible; “invalidity” is an abuse of terminology.

[ladyjane]

sorry
euclids 5th was shown to be wrong

No, it was shown that there are useful geometries which do not use it.

sorry wrong
our reality is not euclidian it is reiman
euclidian only appear to work because it is over small distances
but in fact the geometry of our reality is reiman so euclid 5th is wrong in our reality

which confirms colin leslie deans argument that godels incompleteness theorem is meaningless babble as he cant tell us what true statements are

The proof itself does not need “truth” at all: it stands on its own. Please learn to distinguish proofs from their interpretations.

godel has given a proof - he distinquishes between proof and truth
but we dont know if it is true as he and other mathematicians cant tell us what a true statement is- thus the proof is meaningless babble

I’m really not sure what that interpretation is getting at. There’s no actual induction going on in a proof by mathematical induction.

then read these words

But this is itself induction. And hence the argument is a vicious circle.

so
godele is invalid for 3 reasons

1. he cant tell us wat true statements are
2. he uses the invalid axiom of reducibility
3. he uses circular axioms -and cicularity is considered invalid by text books on logic

[Carleas]

1. He doesn’t need to. He just needs to use the agreed-upon rules for manipulating statements that are assumed to be true.
2. Invalid as an axiom is not the same as invalid or untrue as a statement. If the AR is not true, math is incomplete, which is one of the possible outcomes of Godel’s proof (it is called the “Incompleteness Theorem” after all).
3. Godel tries to create circularity by taking as true everything necessary to make math complete. If you reject his givens, you simply claim that math is incomplete. You do not prove him wrong.

[ladyjane]

1. He doesn’t need to. He just needs to use the agreed-upon rules for manipulating statements that are assumed to be true.

rubbish you cant assume anything is true unles you tell us what true is
and godel cant do this as mathematicians cant
thus his proof is meaningless babble

2. Invalid as an axiom is not the same as invalid or untrue as a statement. If the AR is not true, math is incomplete, which is one of the possible outcomes of Godel’s proof (it is called the “Incompleteness Theorem” after all

rubbish
any proof done with an invalid premiss is logically invalid
you only say this crap because others have so called proved godel correct

if i gave a proof that godel was wrong based on the axiom 1+1= 5 you would say the proof was invalid

3. Godel tries to create circularity by taking as true everything necessary to make math complete. If you reject his givens, you simply claim that math is incomplete. You do not prove him wrong.

if maths is incomplete godel did not prove that because

godele is invalid for 3 reasons

1. he cant tell us wat true statements are
2. he uses the invalid axiom of reducibility
3. he uses circular axioms -and cicularity is considered invalid by text books on logic

[Mr_Reasonable]

Here something for ya…

I think that its true that “1=1” means that if there are two things that are identical to one another, then those things are equal.

Instead of going on with some theoretical nonsense, just tell me why the above statement isn’t true.

And don’t say it’s because of Godel!!!

[ladyjane]

I think that its true that “1=1” means that if there are two things that are identical to one another, then those things are equal.

all u have said is they are =
but you have not said why 1=1 is true
until u can 1=2 could be true

[Mr_Reasonable]

Please elaborate further so that I can grasp this concept.

I don’t feel like you’ve shown my statement to be false.

Explain it to me like I’m in 1st grade.