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Single Idea 16343
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
]
Full Idea
The global reflection principle ∀x(Sent(x) ∧ Bew[PA](x) → Tx) …seems to be the full statement of the soundness claim for Peano arithmetic, as it expresses that all theorems of Peano arithmetic are true.
Clarification
Bew[PA] means provable in Peano arithmetic. T means 'true'
Gist of Idea
The global reflection principle seems to express the soundness of Peano Arithmetic
Source
Volker Halbach (Axiomatic Theories of Truth [2011], 22.1)
Book Ref
Halbach,Volker: 'Axiomatic Theories of Truth' [CUP 2011], p.323
A Reaction
That is, an extra principle must be introduced to express the soundness. PA is, of course, not complete.
Related Idea
Idea 16342
You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system [Halbach]
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]
|
13862
|
There are five Peano axioms, which can be expressed informally
[Wright,C]
|
17853
|
Number truths are said to be the consequence of PA - but it needs semantic consequence
[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]
|