/lit/ - Literature

>>13889678

>>10555183
You know Bertrand, I'm quite fond of your work dismantling the so called "naive set theory." You'll be happy to note that I've found some holes in your "naive PM 'formalism' " and I'll be interested to hear your thoughts on it!

>>10474223
Bert, great to see you old friend! I'm glad to see you've enjoyed my theories, it's always nice when someone is willing to admit their entire worldview is naive. You did it for Frege, I did it for you! Perhaps you'll admit your atheism is equally naive, but I suppose we'll see! Anywho, I must be off, Adele is at home cooking my meal, I don't know what I would do without her! God Bless you Bert, and Happy New Year! Oh sorry, Happy New Year is sufficient for you I suppose, you old grump!

In the other "Godel" thread questions came up concerning the application of the Incompleteness Theorems to Philosophical Systems. I decided to do some research to see if the theorems apply only to arithmetical systems, or if they can be extended. I figure it deserves its own thread. The main questions asked in the thread was "what constitutes a sufficiently complex system," and "does the system have to be arithmetical."

All definitions come from the following: https://www.karlin.mff.cuni.cz/~krajicek/smith.pdf

The formal term is "sufficiently strong." A formal theory of arithmetic T is sufficiently strong if it captures all decidable numerical properties. An example of a decidable numerical property is that a number is prime.

In this particular text, properties are considered equivalent if they have the same extension, i.e. if the set of numbers satisfying them is equivalent. So "square root of 4" and "even prime" are both numerical properties which have the extension "2", so under the above definitions the "square root of two" is the same numerical property as "even prime." The only other criterion for something being a property is that it can be written in the language of your formal theory.

This immediately creates some concern for philosophers seeking to exempt themselves from Godel's findings. Suppose you take a reduced English language as your formal language. You could encode a set of objects in your language with the natural numbers, and then objects which satisfied your property would be the extension of that property. For instance, if we encode every single living creature on the face of the Earth at present with a natural number, this would (as far as I know) be finite. If we define "mammalian" as a property, a subset of living creatures would satisfy that property, and thus a subset of the natural numbers would possess this property.

This allows us to extend Godel's theorems beyond Arithmetical Logic to any Logic whose language can be encoded by the natural numbers. The question then is not if "non-numeric" properties can be considered, but if these properties are decidable. I.e. If we can "decide" if an object possesses that property. This is probably, if you don't want the theorems to effect philosophy, what you want to argue against.

Thus, Godel's Incompleteness theorems apply to a philosophical system if:
1. We can, using an algorithm and with infinite time, decide whether or not that property is true for every possible object in your language. This excludes more candidates for a property than one might think. For instance, the property "encodes a theorem in the 'nice' formal Theory T" does not count as a decidable property.
2. Have a list of axioms in the system which, given infinite time and paper, we could list out

I'd be interested to see if anyone with a more formal education in Logic has any comments. I'm also curios if you guys think "possesses a human consciousness" is a decidable property.

>>10460666
Burt! Another chance encounter! Oh? Your book is selling well, good to hear! Too bad I proved it's incomplete, but it's a useful exercise nonetheless. Anyways, I'll read your musings about a teapot if you'll read my modal proof for the existence of a necessary being; I'm worried about your soul old friend. What? You don't even want to read that crap? Well that's disappointing Burt, I hope you have a nice day though.

>>10437587
Oh heya Bert, I was just putting the finishing touches on this ontological proof for the existence of God. By the way, no hard feelings about me disproving your three volume system of mathematics right? I appreciate your tenacity of course, but I guess every consistent system needs a little faith!

>Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Leibniz rather than Spinoza."
can anyone explain the fundamental difference of Spinoza and Leibniz, especially Leibniz to read his books

>>10331835
>proves your framework incomplete
Everyone needs a little faith Bert! That's why I use my intellect to make ontological proofs for God, and why your work will be remembered as nice but ultimately futile.

Where do I start with logic?
>the Greeks
Is there a chart or something?

>>8950294
not posting the best analytical qt

Curt Gurdle

>tfw I went to a state school in the midwest
>tfw it wasn't even my states flagship state school

Was mathematics created or discovered?

>>6801014

>yeah about that complete and consistent logic

>https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

>Age: 23
>University: University of Wisconsin (Too embarrassed to say which school)
>Highest Level of Education..Major/Minor: B.A. Computer Science with Mathematics as a minor.
>Current Job: Software Engineer
>Literary ambitions: Finish a novel I am proud of.

>>6517269

Am I a pleb if I write a screenplay instead of a novel?

What are the philosophical implications of his incompleteness theory? Any good books on this dude?

My CS professor tells me Gödel's incompleteness theorem does apply to things like philosophy and law (both formal systems) and not just math.

You guys are full of shit.

>>5943899
lol

>un-capitalize that T in truth when you talk about math

>learning an entire language just for a few books

That ain't logical.

Did this guy end mathematics?

>>5027037

>what say you.
>how Russel got his shit pushed in by my incompleteness therum
>oh and Wittgenstein was super gay
>nothing wrong with that
>also one of only a few philosophers to see battle
>nobody dumps all over your ideas about truth in mathematics like Kurt Godel.