Difference between revisions of "Inductive set"
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(Created page with "{{Stub page|grade=A*|msg=Needs another reference}} __TOC__ ==Definition== Let {{M|I}} be a set. We call {{M|I}} an ''inductive set'' if{{rITSTHJ}} both of the following pr...") |
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#* {{M|S(x)}} denotes the [[successor set]] of {{M|X}} | #* {{M|S(x)}} denotes the [[successor set]] of {{M|X}} | ||
{{Caveat|Note that this certainly describes the natural numbers as we require {{M|\emptyset\in I}}, so they're in there. The problem is that rule 2 seems to require that for every element {{M|n}} that {{M|n\cup\{n\} }} is in there too.}} - this seems to be intended{{rSTTJ}} | {{Caveat|Note that this certainly describes the natural numbers as we require {{M|\emptyset\in I}}, so they're in there. The problem is that rule 2 seems to require that for every element {{M|n}} that {{M|n\cup\{n\} }} is in there too.}} - this seems to be intended{{rSTTJ}} | ||
| + | ==See also== | ||
| + | * [[The natural numbers]] | ||
| + | * [[The axiom of infinity]] - positing that an inductive set exists. | ||
==References== | ==References== | ||
<references/> | <references/> | ||
| + | {{Definition|Set Theory}} | ||
Latest revision as of 15:56, 3 February 2017
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Contents
[hide]Definition
Let I be a set. We call I an inductive set if[1] both of the following properties hold:
- ∅∈I - often written 0∈I as 0 is represented by the ∅ - and
- ∀n[n∈I⟹S(n)∈I] - often written as "if n∈I then (n+1)∈I"
- S(x) denotes the successor set of X
Caveat:Note that this certainly describes the natural numbers as we require ∅∈I, so they're in there. The problem is that rule 2 seems to require that for every element n that n∪{n} is in there too. - this seems to be intended[2]
See also
- The natural numbers
- The axiom of infinity - positing that an inductive set exists.
