Week 1 Exercises, due 28 September 2021

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Exercises 1.15 Week 1 Exercises, due 28 September 2021

Please indicate whether the following sentences are true, false, or meaningless, and include a short (1 sentence) justification. Each of these is worth 1 point.

1.

\begin{equation*} \NN_0 = \{n \in \NN_0 : (\exists k \in \NN_0)(n=2k)\} \cup \{n \in \NN_0 : (\exists k \in \NN_0)(n=2k+1)\} \end{equation*}

2.

\begin{equation*} \varnothing = \{n \in \NN_0 : (\exists k \in \NN_0)(n=2k)\} \cap \{n \in \NN_0 : (\exists k \in \NN_0)(n=2k+1)\} \end{equation*}

3.

For every \(x \in \varnothing\text{,}\) one has \(0=1\text{.}\)

4.

There exists \(x \in \varnothing \) such that \(0=0\text{.}\)

A set \(X\) is said to be transitive if and only if, for every \(x \in X\text{,}\) one has \(x \subseteq X\text{.}\) (In particular, every element of a transitive set must itself be a set.) A transitive set \(X\) is called an ordinal if and only if it satisfies the following condition:

\begin{equation*} (\forall x \in X)(\forall y\in X)(((x \in y)\wedge\neg(y\in x)\wedge\neg(x=y))\vee(\neg(x\in y)\wedge(y\in x)\wedge\neg(x=y))\vee(\neg(x\in y)\wedge\neg(y\in x)\wedge(x=y))) \end{equation*}

The following exercises are really about getting the definitions and the use of logic correct. Each is worth 2 points.

5.

Prove that not every transitive set is an ordinal.

6.

Prove that every natural number is on ordinal.

7.

Prove that not every ordinal is a natural number.