more on this theme
|
more from this text
Single Idea 18178
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
]
Full Idea
For Zermelo the successor of n is {n} (rather than Von Neumann's successor, which is n U {n}).
Gist of Idea
For Zermelo the successor of n is {n} (rather than n U {n})
Source
report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Naturalism in Mathematics I.2 n8
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.24
A Reaction
I could ask some naive questions about the comparison of these two, but I am too shy about revealing my ignorance.
The
21 ideas
from Ernst Zermelo
17832
|
Zermelo showed that the ZF axioms in 1930 were non-categorical
[Zermelo, by Hallett,M]
|
13028
|
Replacement was added when some advanced theorems seemed to need it
[Zermelo, by Maddy]
|
17626
|
The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers
[Zermelo]
|
13487
|
In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals
[Zermelo, by Hart,WD]
|
13015
|
Zermelo used Foundation to block paradox, but then decided that only Separation was needed
[Zermelo, by Maddy]
|
13486
|
Not every predicate has an extension, but Separation picks the members that satisfy a predicate
[Zermelo, by Hart,WD]
|
13020
|
The Axiom of Separation requires set generation up to one step back from contradiction
[Zermelo, by Maddy]
|
18178
|
For Zermelo the successor of n is {n} (rather than n U {n})
[Zermelo, by Maddy]
|
13027
|
Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets
[Zermelo, by Maddy]
|
9627
|
Different versions of set theory result in different underlying structures for numbers
[Zermelo, by Brown,JR]
|
13017
|
Zermelo introduced Pairing in 1930, and it seems fairly obvious
[Zermelo, by Maddy]
|
15924
|
Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them?
[Zermelo, by Lavine]
|
10870
|
ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice
[Zermelo, by Clegg]
|
13012
|
Zermelo published his axioms in 1908, to secure a controversial proof
[Zermelo, by Maddy]
|
17609
|
Set theory can be reduced to a few definitions and seven independent axioms
[Zermelo]
|
17608
|
We take set theory as given, and retain everything valuable, while avoiding contradictions
[Zermelo]
|
17607
|
Set theory investigates number, order and function, showing logical foundations for mathematics
[Zermelo]
|
17613
|
We should judge principles by the science, not science by some fixed principles
[Zermelo]
|
15897
|
Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed
[Zermelo, by Lavine]
|
9565
|
Zermelo made 'set' and 'member' undefined axioms
[Zermelo, by Chihara]
|
3339
|
For Zermelo's set theory the empty set is zero and the successor of each number is its unit set
[Zermelo, by Blackburn]
|