I think I can answer this one: you get the mappings by using verbal_symbolic skills AND visual_spatial skills.
{1,2,3,4,...}
↕ ↕ ↕ ↕
{1,2,3,4,...}{1,2,3,4,5,...}
↓ ↕ ↕ ↕ ↕
{1,2,3,4,...}I wish to clarify the following:
My answer simply does not fit to your framework, where according to your framework, things have to be reduced into their building-blocks (two sets by themselves, as BenjaminTR put it) as an initial condition in order to say some meaningful thing about them.
jsfisher, my framework doesn't have the meaning you thing it does.
All relationships is not necessarily all of them at-once (you know, ELSE condition is used).
Visual_spatial expression of a given thing is translated into its verbal_symbolic expression. No belief is involved here.
jsfisher, when using only verbal_symbolic reasoning that enables to make things work only by disassemble it into its building-blocks, you are not using a framework that is at least visual_spatial AND verbal_symbolic.
As it is all along this thread, visual_spatial AND verbal_symbolic framework.
Here are concrete examples (without loss of generality):
{1,2,3,4,5,...}
↓ ↕ ↕ ↕ ↕
{1,2,3,4,...}{2,4,6,8,...}
↓ ↕ ↕ ↕
{1,2,3,...}X Y R
F F F
F T T
T F T
T T TX Y R
F F F (X or Y do not hold)
F T T (Y holds)
T F T (X holds)
T T T (X or Y hold)X Y R
F F F (X or Y do not hold){}
{}X Y R
T F T (X holds){2}
↕
{8}X Y R
F T T (Y holds){2}
↓
{}X Y R
T T T (X or Y hold){2,3}
↓ ↕
{1}Great, so now you are at the level of toilet criticism.
None of your claims about cardinality has anything to do with actual versus potential infinity. The distinction between actual and potential infinity is that an actual infinity is a real set that currently has infinitely many members. A potential infinity refers to the output of a process that has no limit to how many things it will produce, yet will always have produced only finitely many things.
{1,2,3,4,5,...}
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{1,2,3,4,...}{1,2,3,4,5,...}
↓ ↕ ↕ ↕ ↕
{1,2,3,4,...}This is verbal_symbolic-only reasoning of the given subject, which can't deal with the verbal_symbolic AND visual_spatial reasoning of the given subject.
{1,2,3,4,5,...}
↓ ↕ ↕ ↕ ↕
{1,2,3,4,...}1_2_3_4_5_...
↕ ↓ ↕ ↓ ↕
1___2___3_...{...4,2,1,3,5,...}
↓ ↓ ↕ ↕ ↕
{1,2,3,...}The illusion of process is in the minds of persons that use verbal_symbolic-only reasoning of the given subject.
{1,2,3,4,5,...}
↓ ↕ ↕ ↕ ↕
{1,2,3,4,...}
jsfisher, you are still missing http://www.internationalskeptics.com/forums/showpost.php?p=9793013&postcount=3406.
(1,2,3,4, ... )
(1,2,3,4,5, ... )
(1,2,3,4,5,6, ...){2,4,6,8,...}
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{1,2,3,...}{2,4,6,8,...}
↓ ↕ ↕ ↕
{1,2,3,...}Your mathematics is no more than verbal_symbolic-only reasoning that can't handle with function with no inverse.
Traditional Mathematics does not handle with
{1,2,3,4,5,...}
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{1,2,3,4,...}{2,4,6,8,...}
↓ ↕ ↕ ↕
{1,2,3,...}(1,2,3,4,5,6,7,8,9,10, ...)
| | ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕
| |(1,2,3,4,5,6,7,8, ...)
| | | | ↕ ↕ ↕ ↕ ↕ ↕
↓ ↓ ↓ ↓(1,2,3,4,5,6, ...)
Exactly, but unlike sets with bounded amount of objects, sets with unbounded amount of objects can be greater than themselves or even smaller than their proper subsets, if a function with no inverse is used, in additional to the traditional function (which must have an inverse form).