In §17 Halmos defines the principle of transfinite induction (PTI) as follows:
suppose that $S$ is a subset of a well ordered set $X$, and suppose that whenever an element $x$ of $X$ is such that the entire initial segment $s(x)$ is included in $S$, then $x$ itself belongs to $S$; the principle of transfinite induction asserts that under these circumstances we must have $S = X$. Equivalently: if the presence in a set of all the strict predecessors of an element always implies the presence of the element itself, then the set must contain everything.
In the next paragraph he says:
The present statement when applied to $\omega$ is easily proved to be equivalent to the principle of mathematical induction
In §12 Halmos defines the principle of mathematical induction (PMI) as a property of $\omega$, that is, of the intersection of all successor sets:
if $S \subset \omega$, if $0 \in S$, and if $n^+ \in S$ whenever $n \in S$, then $S = \omega$
whose property is a consequence of the existence of $\omega$ and therefore a consequence of the axiom of infinity or the existence of a successor set. So in this context, wouldn't it be vacuous to say that both principles are equivalent when proving the principle of mathematical induction implies assuming the existence of a successor set? And therefore the principle of mathematical induction is true regardless of any hypothesis that may be assumed.
I'm missing something?
Best Answer
It's common practice in the literature and in day-to-day (ordinary, outside of logic) informal math-talk to say two (or more) provable statements are 'equivalent' in the non-technical sense that "any can be easily obtained from any other". It happens with some 'foundational' results in real analysis as, say, the Intermediate Value Theorem, the Heine-Borel compactness theorem of the/any interval, and the Bolzano-Weierstrass theorem, or in functional analysis with the open mapping theorem and friends, or in calculus with the inverse function theorem and the implicit function theorem, or in topology with Brouwer's fixed-point theorem and Sperner's lemma
As you noticed, stricly speaking, this would be trivial/meaningless from the point of view of any logic strong enough to contain/prove the scheme
$\varphi \rightarrow (\psi \rightarrow \varphi)$
but sometimes it just happens that the statements refered to as equivalent are ones for which the more reasonable notion of 'equivalence' as "there exists a proper/weaker subtheory which does not prove either statement, but proves the two implications between them" actually applies: most famously the trio consisting of the Axiom of Choice, the Well-Ordering Theorem, and Zorn's Lemma, all not provable in ZF
It also sometimes happen that some such called 'equivalences' turn out to be false in rather precise senses: when formulated in second-order arithmetic both the Intermediate Value Theorem and Sperner's lemma are provable in the weak 'recursive comprehension' theory, while Heine-Borel and Brouwer's FPT require a stricly strong theory ('weak König's lemma'), and Bolzano-Weierstrass require a stronger one still ('arithmetic comprehension') (take a look at Simpson's Subsystems of second order arithmetic for all of this). That these statements can even appear to be equivalent in that informal sense maybe be due to the set theory most people use being too strong to notice any distinction
tldr: people often say stuff that makes intuitive sense to them without any formal backing, sometimes it's possible to investigate such stuff, and it may or may not agree with the informal picture