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Full Idea
The truths of arithmetic are just the true equations involving particular numbers, and universally quantified versions of such equations.
Gist of Idea
The truths of arithmetic are just true equations and their universally quantified versions
Source
Peter Smith (Intro to Gödel's Theorems [2007], 27.7)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.258
A Reaction
Must each equation be universally quantified? Why can't we just universally quantify over the whole system?
| 13155 | If you add one to one, which one becomes two, or do they both become two? [Plato] |
| 9865 | Daily arithmetic counts unequal things, but pure arithmetic equalises them [Plato] |
| 16929 | 7+5 = 12 is not analytic, because no analysis of 7+5 will reveal the concept of 12 [Kant] |
| 17612 | Arithmetic is just the consequence of counting, which is the successor operation [Dedekind] |
| 14441 | The formal laws of arithmetic are the Commutative, the Associative and the Distributive [Russell] |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| 10026 | Arithmetic must allow for the possibility of only a finite total of objects [Hodes] |
| 15714 | 'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan] |
| 15715 | 'Distributive' laws say if you add then multiply, or multiply then add, you get the same result [Kaplan/Kaplan] |
| 21665 | The fundamental theorem of arithmetic is that all numbers are composed uniquely of primes [Hofweber] |