Cantor’s Continuum “Hypothesis”

Aleph-naught: $\aleph_0 = |\mathbb{N}|$ is the smallest infinite cardinal number.

Recall: $\omega$ is the smallest infinite ordinal. The first ordinal after $0, 1, 2, \cdots$.

$$2^{\aleph_0} = |pow(\mathbb{N})| = | [0, 1] | = | \mathbb{R} | = | {0, 1}^{\omega} | > |\mathbb{N} $$

What does it mean to say it is proven to not be possible to settle the question of whether $\aleph_1 = 2^{\aleph_0}$ with the ZFC axioms?

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On Computable Numbers

The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. (Alan Turing, On Computable Numbers, with an Application to the Entscheidungsproblem, 1936.)

Is $\tau$ computable (by Turing’s definition)? ($\tau = \frac{\text{Circumference of circle}}{\text{radius of circle}}$)

What are the problems with using the State Machine to model computation: $$M = (S, G \subseteq S \times S, q_0 \in S)$$

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What was Turing attempting to model in defining what we know call a Turing Machine?

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$$ TM = (S, T \subseteq S \times \Gamma \rightarrow S \times \Gamma \times \text{\em dir}, q0 \in S, q{Accept} \subseteq S) $$

$S$ is a finite set (the “in-the-head” processing states)
$\Gamma$ is a finite set (symbols that can be writte on the tape)
$\text{\em dir} = { \text{\bf Left}, \text{\bf Right}, \text{\bf Halt} }$ is the direction to move on the tape.

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What is the cardinality of the set of all Turing Machines?

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Prove that there are numbers that cannot be output by any Turing Machine.

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