> > If a complete description of arithmetical truth is not possible, what > > exactly are we talking about? > > We, humans, have a rather good intuition of what is a true > arithmetical sentence, independently of the fact that we have to > recognize that it can be quite tricky to decide if this or that > arithmetical proposition is true or not.
But if arithmetical truth can not be described then what are we talking about? What our intuition tells us? I'm still unclear as to why a complete description of reality is not possible because we apparently can know what we're talking about when we talk about arithmetical truth, despite arithmetical truth not being recursively enumerable. How does one even prove that "arithmetical truth is not recursively enumerable" if one does not have a complete description of "arithmetical truth." Take whatever level of description of arithmetical truth necessary to prove that statement and analogously take "reality" to be described to that level, then it no longer seems your implausibility argument based on arithmetical truth not being recursively enumerable is applicable. What I mean is that couldn't we make statements about a complete description of reality such as it is not recursively enumerable? How does that mean a complete description of reality does not exist, as you said? There is a complete description of reality, though the most evident one is infinite in length, and not recursively enumerable (since arithmetical truth isn't). What I'm after is analogous to a finite basis for a vector space over an infinite field. Can any infinite TOE be abridged into a finite document that, when provided with how to expand the finite document (like the span of a basis in my analogy), is the original infinite document? If so, that would entail that reality is somehow self-similar or, in a sense, holographic in nature. Or if there is an infinite TOE whose number of symbols is of a certain cardinality, what is the smallest cardinal that that TOE can be compressed into? IF that is provably finite then I think that would be interesting. > > What is the outside view of the ultimate reality? > > Assuming mechanism, arithmetical truth is enough. It is infinite and > non algorithmically generable. > It is just absolutely undecidable that there is anything more needed > for the outside view. So by Occam razor ... > > If we are "machine" (survive at some substitution for some level, or > "are fully turing emulable" with "fully" meaning that consciousness is > preserved) then the inside "knowledge" view of arithmetical truth will > be FAR BIGGER than arithmetical truth. Indeed, it corresponds > plausibly to George Levy first person plenitude. That things is much > less definable (if I can say) than arithmetical truth. There is here > an analog of Skolem paradox, with the existence of a structure which > looks small from outside and is immensely big from inside. > > > How is it possible > > to have an outside view of the ultimate reality? > > By assuming mechanism. If mechanism fails the UDA test, we will know > that arithmetical truth is too big or to small. Plausibly too small > (for the reason that we can already conceive, apparently, arithmetical > truth (but an ultrafinitist could contest this). > > Church thesis rehabilitates the Pythagorean Neoplatonist theology. The > relation between the numbers (the true arithmetical relation or > propositions) defines, by the first person indeterminacy, a > differentiating flux of consciousness, which, by the non awareness of > the basic UD execution time (defined by those relations), filters *in > the limit* the possible and sharable infinite consistent histories. > The reason why you are conscious here and now is rather easy. It > depends only on the truth of one sigma_1 arithmetical sentence > (asserting the existence of your computational state relatively to > some universal number). But there are infinitely many equivalent > sentences, both with the same universal numbers and with different > universal numbers. Most relations involve pretty big numbers. > What the first person indeterminacy makes complex is the relation > between 'here and now', and 'there and then'. This, unfortunately > depends on that infinity of universal (among others) numbers, and all > those sigma_1 propositions, and actually all their proofs, which > define the relative measure. > Below your substitution level, an infinity of universal number > compete, for proving your existence, together with all random oracles. > Fortunately, the constraints of self-referential correctness for the > Löbian observers are enough to shed some light on this, and to make > quite plausible that the bottom physical reality will appear linear > and extremely symmetrical, then self-observation break the symmetries > from the point of view of the self-observing machines. > You can see this as Everett spirit applied to arithmetic. Everett > embeds the physicist in the quantum reality. DM embeds the > mathematician in the arithmetical reality. > I am really not seeing how one can have an outside view of the ultimate reality based on what you've said. Can you explain it in more simple terms? If there were an outside to ultimate, then it wasn't ultimate to start with. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.