Set Theory – Is Pigeonhole Principle the Negation of Dedekind-Infinite?

Question

From Wiki, "The Pigeonhole Principle":

In mathematics, the pigeonhole principle states that if n items are
put into m pigeonholes with n > m, then at least one pigeonhole must
contain more than one item.

From "The Mathematical Infinite as a Matter of Method," by Akihiro Kanamori:

In 1872 Dedekind was putting together Was sind und was sollen die
Zahlen?, and he would be the first to define infinite set, with the
definition being a set for which there is a one-to-one correspondence (bijection?)
with a proper subset. This is just the negation of the Pigeonhole
Principle. Dedekind in effect had inverted a negative aspect of finite
cardinality into a positive existence definition of the infinite.

How can this be? Dedekind's definition of infinite makes no reference to natural numbers or any kind of ordering:

A set $X$ is Dedekind-infinite iff there exists a proper subset $X'$ of $X$ and bijection $f:X'\rightarrow X$

See follow-up in my answer below.

Answer 1

Here's Dedekind's definition:

A set $S$ is said to be infinite if there is an injection from $S$ to one of its proper subsets.

There is a hidden other half implied by this definition, namely that a finite set is one that is not infinite. One might imagine that it also says:

A set $S$ is said to be finite if there is no injection from $S$ to any of its proper subsets.

Or equivalently:

There is no injection from a finite set $S$ to one of its proper subsets.

Or equivalently:

If $f$ is a mapping from a finite set $S$ to one of its proper subsets, then $f$ is not an injection

Replacing "injection" with its definition:

If $f$ is a mapping from a finite set $S$ to one of its proper subsets, there are distinct $x$ and $y$ in $S$ with $f(x) = f(y)$

Now comes what I think is the only leap: imagine that the elements of $S$ are the pigeons, and that the pigeonholes are also labeled by elements of $S$. But not every possible element of $S$ is a label, so that the set of labels is a proper subset of $S$. Then the function $f$ says, for each pigeon, which hole it goes into:

If $f$ is a mapping from a finite set of pigeons $S$ to a set of pigeonholes labeled with elements of a proper subset of $S$,there are distinct pigeons $x$ and $y$ with $f(x) = f(y)$

Or equivalently:

If $f$ sends pigeons from a finite set $S$ to a set of pigeonholes labeled with elements of a proper subset of $S$, there are distinct pigeons $x$ and $y$ sent to the same hole

I think this is the version of the pigeonhole principle that Kanamori was thinking of.

There's still a piece missing before we get to your version of the pigeonhole principle, which is that finite sets have sizes, which are numbers. To do this properly is a little bit technical. We define a number to be one of the sets $0=\varnothing, 1=\{0\}, 2=\{0, 1\}, 3=\{0, 1, 2\}, \ldots$. We can define $<$ to be the same as $\in$, or perhaps the restriction of $\in$ to the set of numbers. For example, $1<3$ because $1\in 3 = \{0,1,2\}$.

Then we can show that for any finite set $S$ there is exactly one number $n$ for which there is a bijection $c:S\to n$, and we can say that this unique number is the size $|S|$ of set $S$. Then we can show that if $S$ and $S'$ are finite sets with $S'\subsetneq S$, then $|S'|<|S|$.

Once this "size" machinery is in place, we can transform the statement of the pigeonhole principle above from one about a set and its proper subset to one about the sizes of the two sets:

If $f$ sends pigeons from a set of size $s$ to a set of pigeonholes of size $s'$, with $s'<s$, there are distinct pigeons $x$ and $y$ sent to the same hole

Or, leaving $f$ implicit:

If pigeons are sent from a set of size $s$ to a set of pigeonholes of size $s'$, with $s'<s$, there are distinct pigeons $x$ and $y$ sent to the same hole

Which is pretty much what you said:

If $s$ items are put into $s'$ pigeonholes with $s > s'$, then at least one pigeonhole must contain more than one item.

I hope this is some help.