# Gödel’s 1st theorem is meaningless

#### meaninglessness

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Gödel’s 1st theorem is meaningless
Godels 1st theorem is meaningless

1) Gödel’s 1st theorem

a) “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, p. 250)

note
"... there is an arithmetical statement that is true..."

In other words there are true mathematical statements which cant be proven
But the fact is Godel cant tell us what makes a mathematical statement true thus his theorem is meaningless

#### Michael Larkin

Member
In other words there are true mathematical statements which cant be proven
But the fact is Godel cant tell us what makes a mathematical statement true thus his theorem is meaningless
You might be right until the bolded part. I think you're using the wrong word with meaningless. The theorem does in fact have meaning: it's only that -- if you're right -- one can't use it to say what makes a mathematical statement true.

Truth (and falsity for that matter) is different from meaninglessness. I can think of many statements that are meaningful to millions of people (for example, religion is full of them, as is politics, even to some extent science) notwithstanding the fact that they're quite probably untrue. Indeed, seeing as what is deemed true changes over time, it could be argued that at any one time the vast majority of accepted ideas are untrue. But they're not meaningless in terms of how they influence what people do.

"Meaninglessness" isn't a mathematical concept. Truth/falsity are mathematical concepts. And there are some mathematical statements that haven't as yet been proven and may not, for all we know, ever be. For example, the conjecture that every even number ≥ 6 is the sum of two odd primes.

Empirically, it's been verified for all even numbers up to around 10^40, but beyond that, it hasn't been tested, and in any case one can't go on forever testing every individual even number. We haven't discovered a way of predicting the pattern of primes throughout the number sequence. Prime gaps, for example, can range from 1 in twin primes (e.g. 11, 13 and 71, 73) to the order of a million or more between ginormous known primes, but where they occur can't be predicted: AFAIK, no general formula exists for generating primes and hence prime gaps.

Because we can't make rhyme or reason out of the prime sequence (and may never be able to), we may never be able to prove/disprove the conjecture. It may be an example of what Godel was claiming, and as such, I don't think that's meaningless. One could perhaps, e.g. (mathematicians might have better examples), use the conjecture in generating some kind of encryption algorithm based on even numbers ≤ 10^40, and if so it could hardly be claimed to have no meaning in the real world.

Maybe you ought to pay as much attention to semantics as you seem to to mathematics?

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#### meaninglessness

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Godels theorem

effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true
If Godel said "effectively generated formal theory that proves certain basic arithmetic gibblies, there is an arithmetical statement that is gibbly"
but did not tell us what gibbly or gibblies are/meant you would have no trouble saying hey Godel your statement/ theorem is meaningless

same goes for true maths statement if he cant tell us what makes a maths statement true then his theorem is meaningless

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