Mathematics to be simply an extension of logic

[pwsnafu]

So again, why are these theorems such a big deal? They are massively overhyped.

Mostly historical reasons, it was Hilbert's second problem that started all this. When Russel's paradox came to light, there was panic in the mathematical community that mathematics was going to fall apart. But that was 100 years ago.

You might even have a different meaning for 'incompleteness'. To me incompleteness means that you cannot prove the axioms.

That is not what incompleteness means. A system is complete if every true theorem (i.e. true statements which is not axiom) is provable from the axioms. A good example is Goostein's theorem. Goodstein sequences are clearly sequences of naturals, and yet PA can't prove that they all terminate.

 

[jgens]

Show that Gödel's theorems should give someone who uses a calculator something to worry about.

You really ought to be more specific about which theorems of Gödel you mean. But assuming you mean the incompleteness theorems, then they have little applicability to calculator computations. Then again so does nearly the entirety of mathematics so that really says very little.

I'm also not quite sure that you're using inconsistent in the same way that I do. To me inconsistent means: 'leads to contradictions.'

This is, more or less, the correct notion of inconsistency.

What contradictions do Peano's Axioms lead to?

Probably none because we generally assume that PA is consistent. However the incompleteness theorems tell us the following two things:

1. PA is either incomplete or inconsistent.
2. If PA were consistent, then it could not prove its own consistency.

You seem to be confused about the meaning of incompleteness, so on this point refer to my response below.

You might even have a different meaning for 'incompleteness'. To me incompleteness means that you cannot prove the axioms.

My meaning of incompleteness is the standard one and yours is completely off the mark. What incompleteness roughly means is that there are statements expressible in our axiom schema that are neither provable nor unprovable. Note that within any particular such scheme the axioms are vacuously provable, and as a result, this is very different from not being able to prove axioms. I repeat: If you cannot understand this difference, then do not proselytize about the role of logic in mathematics. If you want the difference explained to you, then start a new thread about the matter.

So again, why are these theorems such a big deal? They are massively overhyped.

I agree that these theorems get over-hyped and as pwsnafu has already said the primary interest in these theorems is historical. What the theorems guarantee, however, is that any set of axioms strong enough to express most of modern mathematics will necessarily have undecidable statements*. So effectively the theorems tell us to stop looking for an all encompassing set of axioms for mathematics and instead just look for something sufficient.

*These undecidable statements are not the axioms of our set theory. Rather they are things like the Whitehead Problem or the Continuum Hypothesis or the existence of large cardinals.

 

[jpz]

Are there any mathematical examples of this?

It's easy; just a google a list of mathematical axioms

 

[jpz]

No it hasn't. All Godel proved was what we already knew: you can't prove axioms. Many axioms are so obvious that no sane person would doubt them. The fact that you can't prove axioms doesn't matter. We can still carry on with the business of deriving highly instructive consequences from a set of axioms.

No, Godel proved there are an infinite number of mathematical axioms, and they are not derivable through any axiomized form of mathematics. Instead, they rely on some insight/intuitive ability.

Can you give some examples? My guess is that we cannot define them now but we will eventually. You need to prove that they are in principle undefinable with logic which is very hard, especially when you don't even have a definition of logic.

No, you can never define all mathematical axioms in one list. Every time you add an axiom, there will be another one required to prove something about that set. I don't have to prove it; that's what Kurt Godel was for.

 

[robertjford80]

So again, why are these theorems such a big deal? They are massively overhyped.

I agree that these theorems get over-hyped and as pwsnafu has already said the primary interest in these theorems is historical. What the theorems guarantee, however, is that any set of axioms strong enough to express most of modern mathematics will necessarily have undecidable statements*.

What started this debate was someone asserted that Gödel's theorems prove that logicism is false. I objected. I can't remember if you took the counterposition but it seems that we are now in agreement that Gödel's theorems do not falsify logicism.

What incompleteness roughly means is that there are statements expressible in our axiom schema that are neither provable nor unprovable.

How is this different from my assertion that all's Gödel's theorems prove is something that we already knew: 'that you can't prove axioms'?

Note that within any particular such scheme the axioms are vacuously provable, and as a result, this is very different from not being able to prove axioms.

You're going to have to explain how 'vacuously provable' is different from unprovable. Axioms are by definition unprovable. You just assume them for the purposes of making deductions. The axiom, a=b, b=c, therefore a=c is not provable, it is assumed.

 

[robertjford80]

The consensus in this thread (if I am correctly interpreting the snippets I have read) seems to be "no", and I will defer to that.

Why are you deferring to a majority opinion when the proponents cannot even say what mathematics and logic are?

 

[Nick O]

Why are you deferring to a majority opinion when the proponents cannot even say what mathematics and logic are?

Odd, the quote shows a name that is not mine when viewed Tapatalk. I can't say whether it does the same on a web browser.

The key point there was that I had only skimmed the thread, and that I am not qualified to have an opinion on this. I mentioned that I may have misinterpreted the discussion as well.

 

[jgens]

What started this debate was someone asserted that Gödel's theorems prove that logicism is false. I objected. I can't remember if you took the counterposition but it seems that we are now in agreement that Gödel's theorems do not falsify logicism.

We were never in disagreement about the issue. I never took a position on that claim. This is REALLY easy to verify by reading the thread and checking the usernames associated to each post. In the future please do so.

How is this different from my assertion that all's Gödel's theorems prove is something that we already knew: 'that you can't prove axioms'?

If you absolutely insist on hijacking this thread instead of starting your own, then here goes: Once you fix a particular axiom schema for mathematics, the axioms trivially prove themselves, so the theorems of Gödel obviously say something very different than this. One of the things they roughly assert is that (assuming consistency) you can make statements that cannot be proved or disproved from the axioms. Examples of this can be found with the Whitehead Problem and with the existence of large cardinals. These results are not axioms in ZFC and they are undecidable in ZFC. These are the kinds of statements the Incompleteness Theorems refer to.

You're going to have to explain how 'vacuously provable' is different from unprovable.

Vacuously provable means there is a proof for them (within the axiom schema). Unprovable means there no proof or disproof for them. Huge difference.

Axioms are by definition unprovable.

They are unprovable in the abstract yes. Within a fixed system of axioms the axioms verify themselves. The Incompleteness Theorems are in the context of fixed systems of axioms.

Why are you deferring to a majority opinion when the proponents cannot even say what mathematics and logic are?

In the future please do not fill my username into someone else's quote. I could be wrong about this, not being a mod and all, but I am fairly certain that is a fantastic way to get banned.

 

[Evo]

Closed pending moderation.