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The new string has to be placed in the assumed list of Reals. If you place it anywhere, you have to recast the Reals. It's not just that it's new by algorithm,
by thenerdfiles 13y ago
The new string has to be placed in the assumed list of Reals. If you place it anywhere, you have to recast the Reals.
It's not just that it's new by algorithm, but it does not have a deterministic place within the assumed list of Reals.
It's a new "Cantor's Paradise" every time you place that new number. And it does not come into existence by the same procedure used to START generating the list. It's a metanumber -- exception to rule. This is why one has to have a metaphysics of number before one accepts the proof. Why Hilbert et al accept the proof on aesthetic grounds, and a whole class of mathematics on aesthetic grounds. The appropriate response is: That's just another real number.
Naturals, etc START somewhere but you are not forced to reindex with each new number.
- thenerdfiles 13y agoIt's not necessary to believe in all those classes of infinities to see the sui generis nature of the mathematical gesture used to generate the Metanumber. Between any two mathematicians, likely the two will pluck very different Rules from their imaginations or backgrounds to produce some other real number. What is interesting is that we have a new real but also a new list. The lists are contiguous games, not a continuous "paradise." Number is the market wherein we SHARE rules. If we give up rule-sharing, the market itself has no value nor do the numbers (and then all order is defeated).
- thenerdfiles 13y agoWhat is cool to consider is that some algorithms for Metanumber might yield ASCII art. In some cases, we might see glyphs. We first have to agree on where to start the reals, which is simple agreement, but the Metanumber strategy could stir up aesthetic commitments since reals do not include an encoding to direct the list itself. Naturals are encoded to place other naturals.