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

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• ShellWillis
In summary, the principle of omniscience is a principle of classical mathematics that is not valid in constructive mathematics. The idea behind the name 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.
ShellWillis
TL;DR Summary
Delta2
Omniscience principle? Starting with a profound sounding term with no definition! I stopped reading.

Klystron and berkeman
I had the same impression, but I’m still interested

mathman said:
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?

ShellWillis said:
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.

Code:
define p:
print( yes )

case 2: The universe is finite.

Code:
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.

Last edited:
Jarvis323 said:
Can you explain what you think is interesting about it and why you titled the thread next generation set theory?
My thoughts as well...

I can’t say

berkeman
ShellWillis said:
I can’t say
Really? Can't or won't?

Jarvis323 said:
Can you explain what you think is interesting about it and why you titled the thread next generation set theory?
ShellWillis said:
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.

1. What is "Next generation set theory"?

"Next generation set theory" is a branch of mathematics that builds upon traditional set theory by incorporating new concepts and techniques. It aims to address some of the limitations of traditional set theory and provide a more comprehensive framework for understanding mathematical structures.

2. How is "Next generation set theory" different from traditional set theory?

Next generation set theory introduces new axioms and principles that allow for the study of larger and more complex mathematical structures. It also incorporates tools from other branches of mathematics, such as category theory and topology, to provide a more holistic approach to understanding sets and their properties.

3. What are some applications of "Next generation set theory"?

"Next generation set theory" has applications in various fields, including computer science, physics, and philosophy. It can be used to study and analyze large data sets, model complex systems, and provide a foundation for understanding the nature of mathematical truth and reality.

4. What are some current developments in "Next generation set theory"?

There are ongoing efforts to develop new axioms and principles that can further expand the scope of "Next generation set theory". Researchers are also exploring connections between this branch of mathematics and other fields, such as logic and category theory, to deepen our understanding of sets and their properties.

5. How is "Next generation set theory" relevant to the scientific community?

As a foundational branch of mathematics, "Next generation set theory" has implications for various scientific disciplines. It can provide new insights and tools for analyzing complex systems and data sets, as well as help us better understand the nature of mathematical structures and their relationships to the physical world.

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