stands for the Absolute Infinite
, the 'last' or 'biggest' (infinite) number
, the Reflection Principle holds that:
for any conceivable property of ordinals, P, if AI has property P then there is at least one ordinal k which is less than AI and which also has property P.
This is motivated by the thought that if
there were some P
which was uniquely a property of AI
, we could simply define AI
as the ordinal with that property, and then add one
to it, getting a bigger number; in which case its Infinitude would not be quite so Absolute
It follows from the reflection principle that the reflection principle itself either describes an inconceivable property or that there must be lesser ordinals for which the reflection principle is also true.
Suppose k0 is a lesser ordinal for which the
reflection principle is true. Then by the reflection principle, there must be an ordinal, k1, which is less than k0 and for which the reflection principle is also true, and so on, until we end up with the infinite set of ordinals:
... k2, k1, k0 for all
of which the reflection principle must be true. This set is not well-defined since every well-defined set of ordinals must have a least member, and this one doesn't, because we can go on forever generating members and each is by definition less than the previous one.
Consequently, we should not regard the reflection principle as expressing a conceivable property of ordinals (but this is ok, since AI is not really an ordinal number, in the same way that V, the class of all sets, is not really a set.)
However, it is possible to state the reflection principle in a weaker form, the Fixed-Point Reflection Principle, which gives more consistent consequences. Paul Mahlo did this in order to arrive at the Mahlo Cardinals1.
This suggests to me that finding larger and larger infinities is roughly equivalent to the practice of finding stronger and better (and hence more complex, it seems) mathematical substitutes for 'conceivable'.
By analogy with the reflection principle, I've thought of two other principles along the same lines:
The Software Specification Principle
For any specification for a non-trivial software requirement there is a piece of software that matches the specification and yet is lesser and distinct from the software that is actually required.
(This is a sort of corollary
of Hofstadter's Law
The Description Principle
Any systematic description of reality falls short. (There is always a distinct and lesser reality that matches the description.)
(See also: inverse spectrum argument
1. The Mahlo cardinals are bigger than the inaccessible cardinals, the hyperinaccessible cardinals, the super-hyper-inaccessible cardinals and so on, but less than the indescribable cardinals, the ineffable cardinals, the partition cardinals, the Ramsey cardinals, the measurable cardinals, the strongly compact cardinals, the supercompact cardinals and the extendible cardinals (given in increasing order of inconceivability. Node away!) (All these are real infinities, genuinely believed in by serious mathematicians in universities around the world. Do you believe that?)
Terms and information taken from Infinity and the Mind, by Rudy Rucker. The (no doubt unoriginal) speculation and extra 'principles' are (in the form given) my fault.