Gödel

[Anonymous No.4688051]

Hi there /sci/

I asked a friend: "what's the different between a PhD in Mathematics and a large pizza"

And he answered: "The pizza does have a base to be built upon"

Then he started talking about mathematical foundations (not sure if it's the proper term though) and Russell and Gödel.

I searched the internet, and all the info I can find is for pros and I do not have that level. Can someone explain what Kurt Gödel said in a way "normal" people can understand? And by normal I mean someone with a decent math level but not that pro.

Thanks /sci/

[Anonymous No.4688090]

You mean the main result of Gödel ((In-)Completesstheorems)?
Or do you want to know something about the foundations of mathematics in general, first order logic and set theory, etc.?

[Anonymous No.4688102]

I mean, why were they so concerned about basing the mathemathics on some "true" axioms and why the classical axioms (Euclid and such) were not valid for them.

[Anonymous No.4688107]

Einstein was a nazi

[Anonymous No.4688147]

Good god, Einstein was Gödel's friend. Also, that is called fallacy.

[Anonymous No.4688164]

The Russel paradox is the observation, that you can't just define any set you like in set theory.

The set A of all sets which doesn't contain itself is a problem, because the question if A contains itself leads to a paradox.

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The main statement of Gödel is that there in strong mathematical theories/frameworks (arithmetic is enough), there are statements, which are true (in a certain sense) in that theory, but which can not be proven.

So it is a statment about something deductive and it's relations to itself.

It's besically a version of the following statment for mathematics:
"This sentence is true, but Emma Stone is the only person in the world, who cannot consistently believe it and convincingly say it."

Another version is to consider a program, which does the following: It copies its own program code into a simulator and then simulates itself and monitors if it comes to an end. The job of the program is to check if the program does never come to an end. But it can obviously never tell you that it does never come to an end (because that would imply it's done). And so it doesn't come to an end after all. But the point is, that it can't tell you that.

The Gödel version is the same idea for strong mathematical theories: There are things which are true from outside, but once the system is strong enough, it's not strong enough to prove these results.

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250).

[Anonymous No.4688219]

Gosh, that was one good explanation!
Thanks god for giving 4chan the ilumination to create /sci/

[Anonymous No.4688326]

<span class="math">[/spoiler]

[Anonymous No.4688346]

>I asked a friend: "what's the different between a PhD in Mathematics and a large pizza"

Did you have a joke set up or something?

[Anonymous No.4688414]

Actually I read the joke in /sci/ and the answer was something like: "with the pizza you can feed your family"

[Anonymous No.4688658]

The only thing I remember about Gödel is that he was the first (maybe not the first) to recognize that time travel was possible via Einsteins equations. This however required a spinning universe.

[Anonymous No.4688677]

>with a decent math level but not that pro.

So what... A bachelors?