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Single Idea 17882

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic ]

Full Idea

It is a remarkable fact that all the arithmetical properties of the natural numbers can be derived from such a small number of assumptions (as the Peano Axioms).

Gist of Idea

It is remarkable that all natural number arithmetic derives from just the Peano Axioms

Source

Michael Potter (Set Theory and Its Philosophy [2004], 05.2)

Book Ref

Potter,Michael: 'Set Theory and Its Philosophy' [OUP 2004], p.92

A Reaction

If one were to defend essentialism about arithmetic, this would be grist to their mill. I'm just saying.

The 14 ideas from 'Set Theory and Its Philosophy'

10702 Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning [Potter]
10703 Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter]
10704 We can formalize second-order formation rules, but not inference rules [Potter]
13041 Collections have fixed members, but fusions can be carved in innumerable ways [Potter]
10707 Mereology elides the distinction between the cards in a pack and the suits [Potter]
10708 Nowadays we derive our conception of collections from the dependence between them [Potter]
13042 If dependence is well-founded, with no infinite backward chains, this implies substances [Potter]
10709 Priority is a modality, arising from collections and members [Potter]
10712 If set theory didn't found mathematics, it is still needed to count infinite sets [Potter]
10713 Usually the only reason given for accepting the empty set is convenience [Potter]
13043 A relation is a set consisting entirely of ordered pairs [Potter]
13044 Infinity: There is at least one limit level [Potter]
17882 It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter]
13546 The 'limitation of size' principles say whether properties collectivise depends on the number of objects [Potter]