Question about induction to infinity with regard to Bolzano's philosophy

Question

I'm a philosophy and mathematics student, and I'm writing a paper on a proof put forward by Bolzano that if we can know one thing to be true, then we can know infinite truths. Put simply, he states that if we know A to be true then the statement "A is true" is true, and the statement "'A is true', is true" is true, and so on. (There are a lot of problems with this but for now lets just focus on the maths of it) Obviously this is a proof by induction. Russell writes about the inductive numbers as a subset of the natural numbers, and how we can't use induction to infinity, because each number the statement is true for must have a predecessor and successor, which are indefinable for ∞. Can someone explain this further, and possibly more correctly for me? Thanks!

Answer 1

Mathematical induction per se cannot prove the existence of an infinite set, like the set $\mathbb N$ of natural numbers.

Usually, induction is assumed as an axiom "defining the essence" of our "intuitive" undertsanding of the succession of natural numbers.

The principle of mathematical induction is stated, in formal way, as :

if $P(0)$ holds and, for any $n$, if $P(n)$ holds, then $P(n+1)$ holds, then $P(n)$ holds for any $n$.

The key-point is the "operation" $+1$ which produce, for any number $n$, its successor $n+1$.

The weak-point of a proof like Bolzano's one (at least in his simplest form) is that induction is not enough to ensure infinity.

For clerness, call $s(x)$ the successor function and consider a simple "universe" $U$ with only two objects : $U = \{ 0,1 \}$ and define the successor as follows :

$s(0)=1$ and $s(1)=0$.

Assume now a property $P$ such that : $P(0)$ holds and that : for any $n$, if $P(n)$ holds, then also $P(s(n))$ holds.

Mathematical induction licenses us to conclude that $P(n)$ holds for any $n$, i.e. that both $P(0)$ and $P(1)$ hold.

But our "universe" $U$ is not infinite.

Answer 2

There are many ways of saying the same thing. Saying it in many ways does not express a new truth each time. So, yes, the number of ways of saying something is without theoretical bound; but, no, this "count" does not represent more than one truth.

Saying "$A$ is true", rather than saying what $A$ represents---a possibly long and complex statement---is a linguistic convention: a short sentence replaces a long one. In ordinary speech, rather than saying "$A$ is true", we would more likely say "it is true", "that is true", "what XY says is true", and so on, as long as the reference is clear from the context; we do not say just "it", "that", etc., because these are not sentences.