A key method in the usual proofs of the first incompleteness theorem is the arithmetization of the formal language, or Gödel numbering: certain natural numbers. Gödel Number. DOWNLOAD Mathematica Notebook. Turing machines are defined by sets of rules that operate on four parameters: (state, tape cell color. Gödel’s numbering system is a way of representing any sentence of the formal language as a number. That means that every sentence of the formal language.
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Gödel Number — from Wolfram MathWorld
I don’t have a mathematical background, i’m just a curious guy so it may be a silly question, i feel like their is a difference between “real” numbers, whatever they may be, and their representation, and i’m getting confused about what is manipulated, using what system, etc. You’re right to sweat the details on this: The entire proof is about the boundary between what can be said in a formal system, what can be proved in it, and the outside world of abstract entities which it attempts to characterize.
So it’s essential to maintain clarity about these things, and draw a sharp distinction between strings in the formal system and what they denote or evaluate to. A Godel numbering is a particular way of assigning numbers to terms and formulas of a particular formal system, provided the system contains enough arithmetic to make that possible.
The integer assigned to a formula is its Godel number. Finally, a note on terminology: The language is the first-order language of arithmetic. Thus, a term e.
The formulas of a formal system in outward appearance are finite sequences of primitive signs variables, logical constants, and parentheses or punctuation dotsand it is easy to state with complete precision which sequences of primitive goel are meaningful formulas and which are not.
Similarly, proofs, from a formal point of view, are nothing but finite sequences of formulas with certain specifiable properties. Of course, for metamathematical considerations it does not matter what objects are chosen as primitive signs, and we numvering assign natural numbers to this use. Consequently, a formula win be a finite sequence of natural numbers, and a proof array a finite sequence of finite sequences of natural numbers. The metamathematical notions propositions thus become notions propositions about natural numbers or sequences of them; therefore they can at least in part be expressed by the symbols of the system itself.
Home Questions Tags Users Unanswered. Like are we manipulating genuine natural numbers or their representation in the system at hand?
I heard about models and stuff but i may be mixing up everything. Does that make any difference? So, you’re reading Godel’s original paper — where? In his collected works? In van Heijenoort’s anthology From Frege to Godel? I think that’s commendable.
Godel is very readable. I am curious, i think everybody could and should be able to understand great mathematical and scientific work. I’m just “another” computer programmer with not much of a college math or serious computer science background, but i’m craving for that knowledge.
I think A LOT of people are too, even the “average Joe” why are documentary so successful if that was not the case. I hope soon i understand every detail of it, make it intuitive and acessible and share it to ANYone.
It goxel like reading and writing. For a long time it was reserved to an elite. Now it’s a given in our modern society for anyone to be able to read and write one another, exchange ideas. It should be the same for those great scientific works. I got excited on reading the intro to the first — that he would be translating the paper, modernizing notation and providing commentary.
My spirits sagged when I saw his first few comments: Unfortunately “improvable” is already taken in English and means something quite different. The next link isn’t as easy on the eyes but I it offers nimbering commentary and I prefer it. Collecting my comments into one answer. So if i’m right, the 45 functions and afterwards always manipulate so called “numerals”, i.
And about the notation: The 45 functions are defined within the system, and others are then derived from them within the system by composition, via substitution of terms for variables. That’s all carried out within the godsl system, yes.
Now it seems more clear in my head. I will read the paper once again, try to find results myself and see if numberjng all clicks. The language is made of symbols: Elliott Mendelson, Introduction to mathematical logic 4th ed -page on. For a complete introduction to this topic, see: For a divulgative books without errors see: An incomplete guide to its use and abuse and: The Complete Guide to the Incompleteness Theorem And the codes we talked about in my previous question are numerals like using the notation in [ plato.
Every expression in the language has a “meaning”: When G defines the formula 1: Sign up or log in Sign up using Google. Sign up using Facebook.