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Is it just me, or do a really large amount of proof's in higher level maths involve really elaborate tautological statements?
You know, stuff that essentially boils down to, this thing exists because it does kinda thing.
I'ts hard for me to think of a specific example while I'm writing this up but I'm sure there are a few you can think of.
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>>8909840
yes its troo
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>>8909840
Tbh, math is circular reasoning. It just happens to be extremely useful circular reasoning. There is nothing wrong with that, as long as it is consistent.
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>>8909840
you can eventually formalize every proof into a series of implications that follow from basic axioms, that's the point of zfc
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>>8909862
ZFC is not complete and we don't know if it's consistent.
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>>8909840
It's not at all cleat what you're asking.
If you mean tautology in the sense of formal logic (which you probably don't), then one may point out that all statements in logic that can be proven are tautologies, i.e. true under all valuations. Stuff like
[math] \neg P \implies (Q \implies Q) [/math]
Other than that, the answer to your question is most likely: no.
>>8909849
It's not circular reasoning.
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>>8909840
Mathematics is a tautology. Each value and operation only has meaning in relation to the others. That doesn't change that it can be used to model and predict the behaviors of the physical world. What's exactly the problem?
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>>8909867
You don't really refute his point.
Even if there are more unprovable statements than provable ones
>you can eventually formalize every proof...
And wether or not the system is inconsistent doesn't affect our capability to just if something is a proof or just a bullshit series of strings. That is to say, even if your system allows you to prove everything, e.g. (1+7)^2=16, you can still prove (1+3)^2=16.
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>>8909867
then your proof is not a proof???????????????
???????????????????????//
??????????????????????????????????
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>>8909840
all math is essentially tautological
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>>8909840
Some proofs regarding existence are constructive while others are non-constructive. For example, the existence of cylic primary decomposition for vector spaces.
In general, constructive proofs are much harder to develop from scratch but more instructive.
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>>8909862
Prove the FTC starting from only the basic ZFC axioms.
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>>8909926
ZFC gives real numbers (dedekind cuts, ect)
And everything follows from there.
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