From Newsgroup: sci.logic
On 09/28/2026 11:22 AM, Ross Finlayson wrote:
On 04/12/2015 01:46 PM, George Greene wrote:
On Sunday, April 12, 2015 at 2:58:19 AM UTC-4, George Greene wrote:
The numeric VERSION of this principle is,
If the NUMBER of pigeons is greater than the NUMBER of holes
(and you put each pigeon in some hole), then there must be at least 1
hole containing at least 2 pigeons.
It's numeric because of the occurrences
of "number", "number", "greater than", 2, and 1.
I guess there must also be a complementary numeric version that if the
number of pigeons is LESS than the number of holes (and, again,
because this is just what happens in real life), you put each pigeon
in a hole, then there must be an empty hole, or, numerically, there
must be at least 1 hole with at most 0 pigeons in it.
I wonder what the -jection version of THAT is?
"If there is no injection on the (whole entire) hole-set into the
pigeon-set, then there is no TOTAL (on the whole of the holes-domain)
surjection on the hole-set onto (the whole of) the pigeon-set"?
"on" gets complicated here. "Onto" is usually NOT complicated but
domain and range ALSO get complicated here. The ZFC
treatment&definition of "functions" by default makes EVERY function
BOTH "total" AND "a surjection".
There can't be any such thing as a partial function because if an
element from the alleged domain-set is not mapped to anything, the
that element BY DEFINITION *IS*NOT*IN* the domain of the function.
Similarly, there is no way for some element of the range-set to be
omitted from the range, because the range IS a set. If some set has
the property that some element of is not the image-under-the-function
of ANY element in the domain, the that element is NOT IN the range of
the function, so you can't say the function is less than surjective on
the basis of its failure to bring THAT element into its range.
You might equally complain that there are no numerical surjections
because all of them fail to get my left big toe into their range.
Surjectivity has to be relative to some set that is PRE-determined
BEFORE the function is evaluated. Ditto for the domain if you are
going to discuss total vs. partial.
This is a topic in its own right, although perhaps a crank one.
This is a good example that modern proof-assistants' fundamental
definitions of "functions" as "total functions" makes for that
they're trivially refutable.
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