Joel David Hamkins on Gödel's Incompleteness, Set-Theoretic Multiverse & Foundations of Mathematics

Prof. Joel David Hamkins is a mathematician and philosopher who is the O'Hara Professor of Logic at the University of Notre Dame. He was also a Professor of Logic at the University of Oxford and Sir Peter Strawson Fellow in Philosophy. Prof. Hamkins authored several books, including Lectures on the Philosophy of Mathematics, Proof and the Art of Mathematics and The Book of Infinity. In this episode, we discuss Gödel, Tarski, set theory and the philosophy of mathematics.

RSam Podcast #31

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[StephenPaulKing]

I am very interested in opinions on the Axiom of anti-foundation and the work of Jon Barwise. Can we imbed ZFC in some version of non-well founded sets?

[BuleriaChk]

Godel expresses wff's in odd numbers
every number is prime relative to its own base n = n(n/n)=n(1_n) (primes do not include division by other numbers)
Goldbach's Conjecture "every even number is the sum of two primes" n + n = 2n
Godel's expression does not include even numbers in his defintion of wff's - they are therefore "undecidable"
(o + e) = o is always odd so is undecidable because of the existence of even numbers (e+e) = e
(o and e are sets of numbers).

Note that the product of differing powers of prime numbers is zero, since the graphs of x^m and x^n only intersect at x = 0 so (x^n)(x^m) = 0. (the reason powers form a basis in polynomial space).

Proof of Fermat"s Theorem for Village Idiots

c = a + b
c^n = [a^n + b^n] + f(a, b, n) (Binomial Expansion)
c^n = a^n + b^n iff f(a, b, n) = 0
f(a, b, n) <> 0
c^n <> a^n + b^n QED
Pythgoras is wrong, Fermat is correct even for n = 2. Someone go tell the physicists (Especially Einstein and Pauli)
and also for multinomials (tell the cosmetologists..)

(Hint: Wiles had to use modular functions, which are only defined on the positive half of the complex plane.)
there are no negative numbers: -c= a-b, b>a iff b-c=a, a >0, a-a = 0, a=a
if there are no negative numbers, there are no square roots of negative numbers. The ""complex" plane is affine to the real plane (1^2 <> 1, sqr(1^2) = 1 <> 2qr(1) (Russsell's Paradox; a number can't both multiply and not multiply itself).

more on this on the physicsdiscussionforum (dot org)