Math Foundations/C

150--

204: Euclid + the failure of prime factorization for z
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A logical problem
Claim:  $$z=10\uparrow\uparrow10+23$$ cannot be factored into prime numbers.
 * Theorem (Unique Factorization) : Every natural number is uniquely factorizable into primes

We are not happy with such a logical inconsistency! What to do? First let us solidify the claim.

So far
A partial prime factorization $$z=3\times13\times139\times163\times18301\times400109\times27997373\times w$$
 * Question: What does $$w$$ look like?

$$\frac{1}{3}=0.\bar{3}$$

$$\frac{1}{3\times39}=\frac{1}{39}\simeq0.\overline{025641}$$

$$\frac{1}{3\times39\times139}=\frac{1}{5421}\simeq0.00018446781036709\ldots$$ ☹️

$$\frac{z}{3\times13\times139}=18446781036709\ldots$$ ☹️

Reciprocals
Set $$m=3\times13\times139=5421$$. Then

$$\phi(m)=2\times12\times138=2^4\times3^2\times23=3312$$

$$10^{2^4\times3^2\times23}\mod m \equiv 1$$

$$10^{2^4\times3^2}\mod m \equiv 2536$$

$$10^{2^4\times3^2\times23}\mod m \equiv 1$$

$$10^{2^4\times23}\mod m \equiv 3337$$

$$10^{2^2\times3\times23}\mod m \equiv 1$$

$$10^{2\times3\times23}\mod m \equiv 1$$

$$10^{3\times23}\mod m \equiv 3613$$

$$\therefore$$

$$ \begin{align*} ord(10) &= 2\times3\times23\\ &= 138 \end{align*} $$ in U(m).

Decimal
$$\frac{z}{3\times13\times39}$$=1844678103670909426305109758347

length of the repeating cycle = 138
 * Exercise 204.1 : How many such repeating cycles are there?
 * Exercise 204.2 : What does the last cycle look like?

All the primes
$$3\times13\times139\times163\times18301\times 400109\times27997373$$

$$ \begin{align*} phi(m) &= 2\times12\times138\times162\times18300\times 400108\times27997372=109989425893460184115200\\ &= 2^11 \cdot 3^7 \cdot 5^2 \cdot 13\cdot 23^2 \cdot 61 \cdot 4349 \cdot 538411 \end{align*} $$

216: The fundamental dream of algebra
https://www.youtube.com/watch?v=QSZsTeO-C1o&ab_channel=InsightsintoMathematics

We can build a foundation

 * It's time to tackle the elephant in the room
 * Modern set theory is a logically inadequate foundation for mathematics
 * Let's carefully examine: the history, the people, the controversies, the sociology ... and the logical structures

Let's Put Set Theory on Trial

 * Set theory is the foundation??
 * Arithmetic is the foundation!
 * Set theory is part of combinatorics
 * Computer science is also part of the foundation

The Underlying Difficulty

 * What is the model for the continuum?
 * The discrete is how we begin
 * Problems arise far to the "right", when numbers get huge
 * The continuous arise from division
 * a.k.a. rational numbers
 * Should we invent irrational numbers?
 * See Famous Math Problems #19

Difficulties Are Not New

 * The ideas of infinity and infinitesimals arose with calculus
 * Calculus worked, so what is the footing of Analysis?
 * Supe up Stevin's decimals into "Real Numbers"

1870's

 * Dedekind, Cantor, and Hilbert gave descriptions of a new number systems
 * Everything depended on Cantor's infinites sets
 * He was very opposed by many mathematicians
 * But with Hilbert's support, he carried the field

Contradictions

 * Hilbert's Formalism and the incorporation of logic into mathematics were tenaciously clung to, despite failing
 * Axiomatic mathematics (ZFC) are modern math (set theory)
 * The foundation was outsourced to logicians