Yes, I understand that it's a part you need in Gödel-numbering, to make the number that holds the logical sentence. Once you have both addition and multiplication, you can do what Gödel did. With Presburger Arithmetic the completeness is lost if you take into account also multiplication:One point though: Godel-numbering is in the meta-theory, but we want to know why we need multiplication in the object theory. But, if I'm not mistaken, we need that it is representable in the object theory; I'd have to study the proof again. — TonesInDeepFreeze
(see On the Decidability ofPresburger Arithmetic Expanded with Powers)adding, for example, the multiplication function x: Z^2 -> Z(or even simply the ‘squaring’ function, from which multiplication is easily recovered) to Presburger arithmetic immediately results in undecidability, thanks to Gödel’s incompleteness theorem
(See here)The issue is not about multiplication per se, or even about the combination of addition and multiplication. The theory of Real Closed Fields has both, and is consistent and complete. The issue is about the strength of induction.
The induction axioms in Presburger arithmetic are the first order approximation of the Peano axiom, and basically do not allow for establishing facts about other facts that have, themselves, to be established inductively.
You cannot get real recursion off the ground unless you have a second order theory of counting, that allows you to represent the sets of integers for with the results are already established.
So to get a first order theory to start doing Gödel's proof, you have to bring in either infinitely many facts about addition, which are needed to establish the relevant results about multiplication, or a few facts about multiplication, itself, as additional axioms.
From the quote, the only difference between recursion and real recursion, that I can think of, is using recursion for reals, in other words recursive real numbers: "A recursive real number may be described intuitively as one for which we can effectively generate as long a decimal expansion as we wish, or equivalently, to which we can effectively find as close a rational approximation as we wish."What do you mean by 'recursion of the reals'? Recursion requires well ordering. — TonesInDeepFreeze
real recursion — ssu
recursive real numbers — ssu
Better deep in knowledge and shallow in misunderstanding. Better deep in love and shallow in hate. — TonesInDeepFreeze
Then you're discussing with the wrong person. — TonesInDeepFreeze
You may prefer whatever you want; there's no need for forgiveness for preferring whatever you like; meanwhile, I prefer to show how you are wrong in saying that 'true' is not defined entirely with the notion of interpretations and not the notion of axioms. — TonesInDeepFreeze
I have some darned fine conversations with myself. — fishfry
The albums 'Conversations With Myself' and 'Further Conversations With Myself'. — TonesInDeepFreeze
Anytime you want jazz album recommendations, just ring. — TonesInDeepFreeze
Jazz is one thing I know a lot about, unlike logic. — TonesInDeepFreeze
unlike logic. — TonesInDeepFreeze
very humble of you. — fishfry
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