Mathematics to be simply an extension of logic

[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.