/sci/ - Science & Math

[Anonymous No.3600658]

Is the statement "Statements can be true" true?

In attempting to prove that it is true, would you not be invoking that which you are attempting to prove in the first place?

Or am I bullshitting myself thinking that this question is as involved as it appears to me?

Pic unrelated.

[Anonymous No.3600669]

loeb's theorem

[Anonymous No.3600673]

pic is always related

[Anonymous No.3600705]

>>3600669

Just by haphazard inspection, it seems that this theorem is implying that mathematics is inductive at its core.

Interesting.

[Anonymous No.3600712]

>>3600705
it's just a crop of the first page of "cartoon guide to loeb's theorem"

good read, it's a fucked up theorem

[Anonymous No.3600717]

>>3600705
Within a set of axioms, it is impossible to prove those axioms to be self-consistent. It was a nightmare for mathematicians of the time when it was proven that you can't prove math is consistent.

[Anonymous No.3600720]

>>3600712
> followup
The theorem is basically,

"If a formal axiomatic system can prove that 'if I can prove P, then P' then P."

[Anonymous No.3600725]

Inductive reasoning is the pride of science and the bane of philosophy.

[Anonymous No.3600732]

>>3600658
lol at OP trying to prove axioms

[Anonymous No.3600736]

Recursion: see recursion.

[Anonymous No.3600738]

Mathematical truth is freedom from contradiction within some axiomatic system. It is well-defined.
Logics tend to be simpler to deal with. Arithmetic's consistency is provably improvable (we must assume it's true as a religious statement), but provable in stronger systems (which themselves are subject to the same provability issue).

Your statement appears philosophical, and I suppose it's true, but statements can be more than true or false in natural languages (paradoxes, "unask the question", ...)
For example "This statement is false" and other stuff having to do with self-recursion.

[Anonymous No.3600742]

>>3600738
> Mathematical truth is freedom from contradiction within some axiomatic system. It is well-defined

http://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem

[Anonymous No.3600750]

>>3600742
But you can encode a computational machine (which itself uses a subset of arithmetic) and have this machine 'reason' about the axioms of some other system (such as itself or some more powerful system).
Either way I didn't mean well-defined in the same system. I meant it in a meta-mathematical sense.

[Anonymous No.3600765]

>>3600750
Yeah, I see that now. Hasty as fuck, sorry.

[Anonymous No.3600800]

I think language can be used to produce an illusion that causes concepts to be perceived as paradoxical and or illogical when the underlying facts remain uninhibited. I think this is one of those, yup OP.

[Anonymous No.3600820]

>>3600658
maybe I'm just tired, but you're confusing causation with an artificial logical construct used to interpret said observation

things don't obey rules just because you say so

[Anonymous No.3600897]

>>3600820
In math they do.

[Anonymous No.3600910]

LISTEN YOU MOTHERFUCKER

THIS BOARD IS ABOUT MATH AND SCIENCE

NOT YOUR FAGGOT ASS RIDDLES AND MIND GAMES

GTFO

[Anonymous No.3601010]

>>3600910

umad?

[Anonymous No.3601137]

>In attempting to prove that it is true, would you not be invoking that which you are attempting to prove in the first place?

Sort of. You would have to "jump up a level" to make statements about statements. And we don't have a system of communicating other than well.. statements. For argument's sake you could decide that "thoughts" are of a higher level than statements. It wouldn't be a real mathematical proof of course.

[Anonymous No.3601398]

http://en.wikipedia.org/wiki/Principle_of_bivalence

short answer is, in bivalent logics, any well-formed formula COULD be true (satisfiability) within that logic

A formula A in a language Q is satisfiable if it is true for some interpretation of Q

modifying your statement to "well formed formulas can be true" makes it true in bivalent logics