> On 25 Feb 2020, at 09:18, Alan Grayson <[email protected]> wrote: > > > > On Monday, February 24, 2020 at 8:13:25 PM UTC-7, Brent wrote: > > > On 2/24/2020 6:28 PM, Alan Grayson wrote: >> >> >> On Monday, February 24, 2020 at 6:24:12 PM UTC-7, Brent wrote: >> >> >> On 2/24/2020 4:50 PM, Alan Grayson wrote: >>> ISTM that Peano's Postulates clearly imply positive and negative integers, >>> zero, and arithmetic. What's the contrary argument? TIA, AG >> >> Contrary to what? >> >> Brent >> >> I'm referring to the argument of those who claim the Peano's Axioms don't >> imply arithmetic (when IMO, they obviously do). How does that argument >> go?AG > > Dunno. How do such arguments go? > > Brent > > Concerning PA, Peano's Axioms, how do we know that a set exists that > satisfies those axioms,
That is the question. With mechanism the answer is very simple and clear: we don’t know. We just hope. But most would say that they believe in a model of arithmetic, because they believe in the structure (N, 0, +, *), learn in secondary school. That is not a proof, because this relies on set theory, which is richer than Peano. Can we prove the existence of a model of set theory? Yes, but only in a still richer theory, like ZF + it exists an inaccessible cardinal. By Gödel’s incompleteness theorem, we know that NO theory at all can prove the existence of a model of itself, and that is why the machine cannot avoid … religion (the belief in a reality satisfying/veufying their beliefs). > or is it just assumed it exists? AG Yes, but not in then theory. Only in the vague and intuitive metattheory. For mechanism, you need just to agree with 1) 0 ≠ s(x) (0 is not the successor of a number) 2) s(x) = s(y) -> x = y (different numbers have different successors) 3) x = 0 v Ey(x = s(y)) (except for 0, all numbers have a predecessor) 4) x+0 = x (if you add zero to a number, you get that number) 5) x+s(y) = s(x+y) (if you add a number x to the successor of a number y, you get the successor of x added to y) 6) x*0=0 (if you multiply a number by 0, you get 0) 7) x*s(y)=(x*y)+x (if you multiply a number x by the successor of y, you get the number x added to the multiplication of the number x with y) The metaphysics and physics are derived from this, with the mechanist hypothesis in the background, or at the metalevel, or as axioms in the mind of some universal numbers/machine. Then, normally, if you remember your math course, you might believe that indeed this is consistant, and that is illustrated by the common belief in the intuitive model (N, 0, +, *) learned in secondary school. Note that the axioms above, or Turing equivalent one, are parst of all theories used in science today. Mechanism entails that the simpler and most common belief are enough, and worst, cannot be completed. The theology and physics are derived in the internal phenomenology of the numbers. Bruno > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/a81a47f3-dda5-4f26-b97f-dd9cb92c7f37%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/a81a47f3-dda5-4f26-b97f-dd9cb92c7f37%40googlegroups.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/A7EF6212-65D7-405F-80B7-49F193C250F1%40ulb.ac.be.
