The Principles of Mathematics (1903)
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Cue Gödel... [1]
[1] https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompletenes...
The second order Peano axioms are a superset of the first order axioms. There is one small change in one of the axioms and that is the difference between the two systems.
When I said "true statements that can't be proven" it should have been qualified to a particular set of axioms. That is, I am claiming each set of axioms has it's own set of true but unprovable statements but none have an empty set of those statements. Correct or not?
Based what you say about the second order axioms it seems not, in which case I have some reading to do :)
The problem with doing this is that the collection of all true statements about the Natural numbers is not recursively enumerable. There is no effective (algorithmic) procedure for determining when a given statement is an axiom or not. This means that under this system we can't find all true statements, even theoretically, using a computer because of the non recursively enumerable nature of the axiom system.
The first order system provides a nice recursively enumerable set of axioms. But these axioms are not powerful enough to deduce all true statements about the Natural numbers. The second order axioms are powerful enough but are not recursively enumerable.
Recursively enumerable was the goal of Hilbert and others because they wanted to reduce mathematics to an algorithmic or mechanical process of verification. That isn't possible and this is the main reason why Roger Penrose (and me) think that computers will never be able to do mathematically what humans can do.
> This collection forms an axiom system for the Natural numbers and all true statements are provable in this system. Indeed, every true statement is an axiom.
You have defined the axiom system as the set of all true statements about natural numbers so of course all true statements are provable! But crucially ...
> The problem with doing this is that the collection of all true statements about the Natural numbers is not recursively enumerable.
We want an extensional definition of a set of axioms. One way to get that is to enumerate it! This is important because ...
> Recursively enumerable was the goal of Hilbert and others because they wanted to reduce mathematics to an algorithmic or mechanical process of verification.
I can sympathize with this goal being that I use a proof assistant quite frequently.
Thank you for taking the time to respond!
I will add one more thing for completeness sake. There are statements that are true of the natural numbers, the natural numbers that you and I think of when see this term, that can't be proven in the first order theory. This means that there is a non-standard model in which there is a non-standard integer that is a counterexample to the statement. The second order axioms are categorical so there is a proof of such a statement using the second order axioms. One may not be able to find such a proof but there is one.
My main point is I disagree with the view of mathematics as nothing more than some axiomatic program-- in 1903 many were hopeful that a system (like Russell's formal logic in Principia) could simply generate the truths of mathematics. Gödel shattered that dream.
I just don't see how this follows from Godel. It gives us a more expansive view of math, but I don't see how any fundamental understanding is overturned. I don't see how this takes away from the connection between axioms and theorems. The characterization of math as discovering the logical consequences of axioms is just as true.
It is absolutely not a philosophy text. It's an attempt at defining a rigorous definition of the basic axioms we take for granted in mathematics in terms of pure logic, and then examining whether those basic rules of logic are themselves irreducible forms of nature or further creations of man. Essentially it seeks to support all higher mathematical thought by not just taking axioms for granted, but formally proving every one.
In my experience, logic falls into two catgeories: using math to study a formal system of logic; and determining "foundations for all math" and determining what is an "irreducible form of nature" and other such prosaic things (can the phrase "such that" be defined?). In my humble opinion, the former is mathematics and the latter philosophy.
i am almost positive all of the views recited in this piece where still considered relevant today have since been made more accessible from textbooks to youtube videos
do you need to, or even need to want to, read this piece by russell? hardly, but at the same time it was written by a well respected individual in the field and there could be value in trying to dissect the text
regardless of your views on any single piece of writing i would wholeheartedly encourage your interest in mathematics
much like how i would encourage someone who dislikes or finds shakespeare too dense to still pursue a career in writing if they so desired but i would still recommend them to take another look