Recently, I became fascinated with Set Theory and I am willing to learn more. Although, there are some aspects that I would like to understand before doing it. A lot of questions concerning the foundations of Mathematics as been posted and I went to read some of them. My question is closely related to Set theoretic concepts in first order logic, to be more precise, in the answer given by Peter Smith.
Let’s (naively) pretend that someone not familiar with mathematical reasoning or sets wants to learn about set theory and we want to teach them. I’m trying to understand how this could be done in the most self contained way (since that person doesn’t have a strong mathematical background).
I think it would be really helpful to learn some logic first. So, I went to search and I picked up the book Logic and Structure by Dirk van Dalen. Although, there are a lot of mathematical terms that “appear” to come from set theory. But this can be solved: just aboard logic “verbally” (as in the link I mentioned).
I was trying to do this but I think this is quite “awkward” to say the least. For example, Dalen starts by presenting the alphabet of propositional logic and show how propositions can be constructed. In his definition, he uses the term smallest set
Definition $2.1.1$ The language of propositional logic has an alphabet consisting of
$(i)$proposition symbols: $p_0, p_1, p_2, \dots$,
$(ii)$connectives: $\wedge$, $\vee$, $\rightarrow$, $\neg$, $\leftrightarrow$, $\bot$,
$(iii)$auxiliary symbols: $($, $)$.
Definition $2.1.2$ The set $\mathsf{PROP}$ of propositions is the smallest set $X$ with the properties
$(i)$$p_i \in X$ ($i \in N$), $\bot \in X$,
$(ii)$$\varphi, \psi \in X \implies (\varphi \wedge \psi), (\varphi \vee \psi), (\varphi \rightarrow \psi), (\varphi \leftrightarrow \psi) \in X$,
$(iii)$$\varphi \in X \implies (\neg \varphi) \in X$.
Someone that has studied (at least naive) set theory will understand what this means, but can one create the same without using this ideas?
What I want to know is, to develop propositional calculus, how can one express the second definition without using the term smallest set?
Because this is quite a good idea to model the set of propositions. It allows us to prove the Induction Principle
Theorem $2.1.3$ (Induction Principle) Let $A$ be a property, then $A(\varphi)$ for all $\varphi \in \mathsf{PROP}$ if
$(i)$$A(p_i)$, for all $i$, and $A(\bot$),
$(ii)$$A(\varphi), A(\psi) \implies A((\varphi \square \psi))$,
$(iii)$$A(\varphi) \implies A((\neg \varphi))$.
This principle is good, for example, to show that every proposition will have one and only one truth value. But how can I prove it without the previous notion of smallest set?
Is there even a way to handle propositional and first order logic without using set theory so then I can use it to work with set theory? If yes, how? (Or where can I find such an approach in the literature?)
Thank you for your attention!