Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Zermelo set theory
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== The aim of Zermelo's paper == The introduction states that the very existence of the discipline of set theory "seems to be threatened by certain contradictions or "antinomies", that can be derived from its principles – principles necessarily governing our thinking, it seems – and to which no entirely satisfactory solution has yet been found". Zermelo is of course referring to the "[[Russell's paradox|Russell antinomy]]". He says he wants to show how the original theory of [[Georg Cantor]] and [[Richard Dedekind]] can be reduced to a few definitions and seven principles or axioms. He says he has ''not'' been able to prove that the axioms are consistent. A non-constructivist argument for their consistency goes as follows. Define ''V''<sub>α</sub> for α one of the [[Ordinal number|ordinals]] 0, 1, 2, ...,ω, ω+1, ω+2,..., ω路2 as follows: * V<sub>0</sub> is the empty set. * For α a successor of the form β+1, ''V''<sub>α</sub> is defined to be the collection of all subsets of ''V''<sub>β</sub>. * For α a limit (e.g. ω, ω路2) then ''V''<sub>α</sub> is defined to be the union of ''V''<sub>β</sub> for β<α. Then the axioms of Zermelo set theory are consistent because they are true in the model ''V''<sub>ω路2</sub>. While a non-constructivist might regard this as a valid argument, a constructivist would probably not: while there are no problems with the construction of the sets up to ''V''<sub>ω</sub>, the construction of ''V''<sub>ω+1</sub> is less clear because one cannot constructively define every subset of ''V''<sub>ω</sub>. This argument can be turned into a valid proof with the addition of a single new axiom of infinity to Zermelo set theory, simply that ''V''<sub>ω路2</sub> ''exists''. This is presumably not convincing for a constructivist, but it shows that the consistency of Zermelo set theory can be proved with a theory which is not very different from Zermelo theory itself, only a little more powerful.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)