Exploring the Next Generation of Set Theory: A Discussion on an Intriguing Paper

[ShellWillis]

TL;DR Summary

A good read needing confirmation

https://www.cs.bham.ac.uk/~mhe/papers/omniscient-journal-revised.pdf

Might be my favorite article I’ve ever came across
I would like to see some interpretations on it to broaden my currently very narrow point of view…

Have fun!
-oliver

 

[mathman]

Omniscience principle? Starting with a profound sounding term with no definition! I stopped reading.

 

[ShellWillis]

I had the same impression, but I’m still interested

 

[Jarvis323]

Omniscience principle? Starting with a profound sounding term with no definition! I stopped reading.

Isn't the definition given in the first sentence of the paper after the abstract?

 

[Jarvis323]

Summary:: A good read needing confirmation

https://www.cs.bham.ac.uk/~mhe/papers/omniscient-journal-revised.pdf

Might be my favorite article I’ve ever came across
I would like to see some interpretations on it to broaden my currently very narrow point of view…

Have fun!
-oliver

Can you explain what you think is interesting about it and why you titled the thread next generation set theory?

The paper is about constructive mathematics.

Constructive mathematics limits what you can say exists to things which you can construct. So it's less powerful than every day mathematics, but it's more intuitive to some people. As an example (based on the slides in the link below).

proposition: There is a program $p$ out there that prints yes if the universe is infinite and no if the universe is finite.

Proof:

case 1: The universe is infinite.

define p:
    print( yes )

case 2: The universe is finite.

define p:
    print( no )

In constructive mathematics this is considered cheating and isn't allowed. And the reason we were able to do it is considered to be stemming from the law of excluded middle, according to the slides, which says every proposition is either true or false.

A constructive proof would have to lead to an actual program that can tell us the answer.

https://home.sandiego.edu/~shulman/papers/rabbithole.pdf

The principle of omniscience is talked about here:

In logic and foundations, a principle of omniscience is any principle of classical mathematics that is not valid in constructive mathematics. The idea behind the name (which is due to Bishop (1967)) is that, if we attempt to extend the computational interpretation of constructive mathematics to incorporate one of these principles, we would have to know something that we cannot compute. The main example is the law of excluded middle (EMEM); to apply p∨¬pp \vee \neg{p} computationally, we must know which of these disjuncts hold; to apply this in all situations, we would have to know everything (hence ‘omniscience’).

https://ncatlab.org/nlab/show/principle+of+omniscience

This is my fuzzy take on what the paper is saying.

The paper is about the existence of some kinds of infinite sets that surprisingly have the property that, for those sets, this particular omniscience principle (in the start of the introduction) is satisfied without breaking constructivism. And that somehow shows that a certain branch of constructive mathematics can be more powerful than previously thought, and the excluded middle, which is avoided or restricted because it can break constructiveness, as in the example, can be used in a less restricted way in a variant of constructive mathematics.

This could be interesting in a practical way for all I know, because there are a lot of important questions in mathematics which may be true, but not have constructive proofs. I'm not sure of the implications this paper has in terms of particular problems beyond what's in the paper. But, take for example the famous question: does $P=NP$? If it is proven true, and the proof is constructive, it could revolutionize computing. If it is proven true, but not with a constructive proof, then the answer is hardly satisfying because we still can't do anything with it.

 

[Mark44]

Can you explain what you think is interesting about it and why you titled the thread next generation set theory?

My thoughts as well...

 

[ShellWillis]

I can’t say

 

[berkeman]

I can’t say

Really? Can't or won't?

 

[berkeman]

Thread is closed for Moderation...

 

[berkeman]

Can you explain what you think is interesting about it and why you titled the thread next generation set theory?

I can’t say

Since the paper under discussion is in a peer-reviewed journal, this discussion is re-opened provisionally. @ShellWillis you really need to do a better job of leading this discussion. Please respond to the key questions that @Jarvis323 has asked, or the thread will likely be closed permanently. Thank you.