/sci/ - Science & Math

>>14632516
>Hegel was right
I hope not. That shit is more schizo and complicated than the very worst stuff I encountered in physics. If that's how the world is I'd rather be wrong.

I got the Phenomenology of Spirit as an audiobook to try to power through it, which has worked for me on some pretty complex stuff, Wilczek's intuitive description of QCD for example, and it was actually painful to listen to someone try to cram that many clauses into one interval of time. I just read a secondary source instead.

>>12736312
nope

>>10820686 #
>diagonalize
The diagonalization construction would give you a tuple of the length of the size domain of the diagonalization function (I.e. length of |w|)
That's why I said don't confuse it with the cardinal expression of exponentiation.

Ordinal w^w doesn't require you to go from w to w. It doesn't hold a tuple with infinite length.
Instead you have tuples of always growing length and always growing quantity of used symbols.

Look at the linked Exponentiation section of the ordinal arithmetic Wikipedia article:
https://en.wikipedia.org/wiki/Ordinal_arithmetic#Exponentiation
There's even ordinals beyond w^w^w^w^... which are still countable.
Because it's not about objects that look like real numbers/infinite floats, instead it's about infinitely deep nestings of counting tuples of tuples.

Related (but not to your question)
https://en.m.wikipedia.org/wiki/Gentzen%27s_consistency_proof

But in relation to the original question, I should point out that there is in principle conceptualised also
https://en.m.wikipedia.org/wiki/Infinitary_logic

>>10820686
>diagonalize
The diagonalization construction would give you a tuple of the length of the site domain of the diagonalization function.
That's why I said don't confuse it with the cardinal expression of exponentiation.
Ordinal w^w doesn't require you to go from w to w. It doesn't hold a tuple with infinite length.
Instead you have tuples of always growing length and always growing quantity of used symbols.

Look at the linked Exponentiation section of the ordinal arithmetic Wikipedia article.
There's even ordinals beyond w^w^w^w^... which are still countable.
It's not about objects that look like real numbers/infinite floats, instead it's about infinitely deep nestings of counting tuples of tuples.

Related (but not to your question)
https://en.m.wikipedia.org/wiki/Gentzen%27s_consistency_proof

>>10527807
How so?
I get that you don't want contradictions, but why would you apriori formulate it to be expressed by some statements that proves EVERYTHING.

To me this way of looking at contradiction is merely an artifact of strong introduction rules in the classical theory.
If 'A' holds, then 'A or B' holds. (or-introduction)
Now if 'not A' holds and 'A or B' holds, B holds (or-elimination)
And B was arbitrary, so now A and not A concluded any B.
Okay, but all this heavily depends on non-relavant rules* for or and if you start with a weaker framework, you'd not conclude this "everything follows" principle

I mean "relevant rules" as in
https://en.wikipedia.org/wiki/Relevance_logic
Because the or-introduction of B was not tied to what A says.

So I don't think it makes sense, except if you have strong binary semantics. Generally, explosion is too brutal and you can speak of contradiction without it.