more from this thinker
|
more from this text
Single Idea 18246
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
]
Full Idea
Dedekind's demonstrations nowhere - not even where he comes to cardinals - involve any property distinguishing numbers from other progressions.
Gist of Idea
Dedekind failed to distinguish the numbers from other progressions
Source
comment on Bertrand Russell (The Principles of Mathematics [1903], p.249) by Stewart Shapiro - Philosophy of Mathematics 5.4
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.175
A Reaction
Shapiro notes that his sounds like Frege's Julius Caesar problem, of ensuring that your definition really does capture a number. Russell is objecting to mathematical structuralism.
The
25 ideas
with the same theme
[set of arithmetic axioms proposed by Dedekind and Peano]:
18096
|
Zero is a member, and all successors; numbers are the intersection of sets satisfying this
[Dedekind, by Bostock]
|
13949
|
All models of Peano axioms are isomorphic, so the models all seem equally good for natural numbers
[Cartwright,R on Peano]
|
18113
|
PA concerns any entities which satisfy the axioms
[Peano, by Bostock]
|
17634
|
Peano axioms not only support arithmetic, but are also fairly obvious
[Peano, by Russell]
|
5897
|
0 is a non-successor number, all successors are numbers, successors can't duplicate, if P(n) and P(n+1) then P(all-n)
[Peano, by Flew]
|
14422
|
Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms
[Russell]
|
14423
|
'0', 'number' and 'successor' cannot be defined by Peano's axioms
[Russell]
|
7530
|
Russell tried to replace Peano's Postulates with the simple idea of 'class'
[Russell, by Monk]
|
18246
|
Dedekind failed to distinguish the numbers from other progressions
[Shapiro on Russell]
|
18097
|
The Peano Axioms describe a unique structure
[Bostock]
|
10068
|
Natural numbers have zero, unique successors, unending, no circling back, and no strays
[Smith,P]
|
16902
|
Peano arithmetic requires grasping 0 as a primitive number
[Burge]
|
17441
|
Wright thinks Hume's Principle is more fundamental to cardinals than the Peano Axioms are
[Wright,C, by Heck]
|
17853
|
Number truths are said to be the consequence of PA - but it needs semantic consequence
[Wright,C]
|
13862
|
There are five Peano axioms, which can be expressed informally
[Wright,C]
|
17854
|
What facts underpin the truths of the Peano axioms?
[Wright,C]
|
10058
|
No two numbers having the same successor relies on the Axiom of Infinity
[Musgrave]
|
17792
|
1st-order PA is only interesting because of results which use 2nd-order PA
[Mayberry]
|
13657
|
First-order arithmetic can't even represent basic number theory
[Shapiro]
|
10202
|
Natural numbers just need an initial object, successors, and an induction principle
[Shapiro]
|
17459
|
Frege's Theorem explains why the numbers satisfy the Peano axioms
[Heck]
|
17882
|
It is remarkable that all natural number arithmetic derives from just the Peano Axioms
[Potter]
|
17887
|
PA is consistent as far as we can accept, and we expand axioms to overcome limitations
[Koellner]
|
16343
|
The global reflection principle seems to express the soundness of Peano Arithmetic
[Halbach]
|
16321
|
The compactness theorem can prove nonstandard models of PA
[Halbach]
|