Subsystems of Second Order Arithmetic (Perspectives in Logic) 2nd Editio

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  Ok. Looks like Reverse Mathematics is the study of proving axioms of _countable_ mathematics and not uncountable mathematics. So anything related to set theory will be that which is countable.450+ pages. Let’s see if I can get through it.

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 The terms and formulas of the language of second order arithmetic are as follows. Numerical terms are number variables, the constant symbols 0 and 1, and t 1 + t 2 and t 1 · t 2 whenever t 1 and t 2 are numerical terms.
Here + and · are binary operation symbols intended to denote addition and multiplication of natural numbers. (Numerical terms are intended to denote natural numbers.) Atomic formulas are t 1 = t 2 , t 1 < t 2 , and t 1 ∈ X where t 1 and t 2 are numerical terms and X is any set variable. (The intended meanings of these respective atomic formulas are that t 1 equals t 2 , t 1 is less than t 2 , and t 1 is an element of X .) Formulas are built up from atomic formulas by means of propositional connectives ∧ , ∨ , ¬ , → , ↔ (and, or, not, implies, if and only if), number quantifiers ∀ n, ∃ n (for all n, there exists n), and set quantifiers ∀ X , ∃ X (for all X , there exists X ).
A sentence is a formula with no free variables.

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 “As is customary in mathematical logic textbooks”? Ok Id better read a basics of mathematical logic first….In writing terms and formulas of L 2 , we shall use parentheses and brack-
ets to indicate grouping, as is customary in mathematical logic textbooks.
We shall also use some obvious abbreviations. For instance, 2 + 2 = 4
stands for (1 + 1) + (1 + 1) = ((1 + 1) + 1) + 1,

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k i’d better read this first:

 https://en.wikipedia.org/wiki/Logicism

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 https://en.wikipedia.org/wiki/Boolean_algebra

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