<div dir="ltr">Looks nice! Note that Michael has done an extensive development of the ordinals in<div><br></div><div> <holdir>/examples/set-theory/hol_sets</div><div><br></div><div>and transfinite induction is in there somewhere.</div><div><br></div><div>Konrad.</div><div><br></div></div><div class="gmail_extra"><br><div class="gmail_quote">On Wed, Jun 29, 2016 at 5:09 PM, Peter Vincent Homeier <span dir="ltr"><<a href="mailto:palantir@trustworthytools.com" target="_blank">palantir@trustworthytools.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr">I've just made my first contribution to CakeML. This consists of two files within the subdirectory "flame/set-theory", which are modified versions of the corresponding files under "candle/set-theory".<div><br></div><div>My primary reference for this work was "Set Theory, The Third Millennium Edition" by Thomas Jech, Springer, 2006.<br><div><br></div><div>These files extend the existing specification of set theory to full ZFC. This includes two axioms that were previously omitted, the Axiom of Regularity and the Axiom Schema of Replacement:</div><div><div><br></div><div><font face="monospace, monospace">val regular_def = Define`</font></div><div><font face="monospace, monospace"> regular ^mem ⇔ ∀x. (∃y. mem y x) ⇒ ∃y. mem y x ∧ ∀z. ~(mem z x ∧ mem z y)`</font></div><div><font face="monospace, monospace"><br></font></div><div><font face="monospace, monospace">val is_functional_def = Define`</font></div><div><font face="monospace, monospace"> is_functional (R:'a -> 'b -> bool) ⇔ ∀x y z. R x y ∧ R x z ⇒ y = z`</font></div><div><font face="monospace, monospace"><br></font></div><div><font face="monospace, monospace">val replacement_def = Define`</font></div><div><font face="monospace, monospace"> replacement ^mem ⇔</font></div><div><font face="monospace, monospace"> ∀R. is_functional R ⇒</font></div><div><font face="monospace, monospace"> ∀d. ∃r. ∀y. mem y r ⇔ ∃x. mem x d ∧ R x y`</font></div></div><div><br></div><div><div><div><font face="monospace, monospace">val is_set_theory_def = Define`</font></div><div><font face="monospace, monospace"> is_set_theory ^mem ⇔</font></div><div><font face="monospace, monospace"> extensional mem ∧</font></div><div><font face="monospace, monospace"> (∃sub. is_separation mem sub) ∧</font></div><div><font face="monospace, monospace"> (∃power. is_power mem power) ∧</font></div><div><font face="monospace, monospace"> (∃union. is_union mem union) ∧</font></div><div><font face="monospace, monospace"> (∃upair. is_upair mem upair) ∧</font></div><div><font face="monospace, monospace"> regular mem ∧</font></div><div><font face="monospace, monospace"> replacement mem`</font></div></div></div><div><font face="monospace, monospace"><br></font></div><div>In addition, I have modified the Axiom of Infinity,</div><div><br></div><div><div><font face="monospace, monospace">val is_infinite_def = Define`</font></div><div><font face="monospace, monospace"> is_infinite ^mem s = INFINITE {a | a <: s}`</font></div><div><br></div><div><div>not really deleting it, but replacing it by the inductive property, taken from page 12 of "Set Theory, The Third Millennium Edition" by Thomas Jech, Springer, 2006, using the successor operator as defined (Definition 1.20) on page 19 of "Introduction to Set Theory" by J. Donald Monk, McGraw Hill, 1969.</div></div><div><div><br></div><div><font face="monospace, monospace">val suc_def = Define`</font></div><div><font face="monospace, monospace"> suc ^mem x = x ∪ Unit x`</font></div><div><font face="monospace, monospace"><br></font></div><div><font face="monospace, monospace">val _ = Parse.overload_on("Suc",``suc ^mem``)</font></div></div><div><font face="monospace, monospace"><br></font></div><div><font face="monospace, monospace">val is_inductive_def = Define`</font></div><div><font face="monospace, monospace"> is_inductive ^mem s ⇔</font></div><div><font face="monospace, monospace"> ∅ <: s ∧ ∀x. x <: s ⇒ Suc x <: s`</font></div></div><div><div><div><font face="monospace, monospace"><br></font></div><div><font face="monospace, monospace">val is_model_def = Define`</font></div><div><font face="monospace, monospace"> is_model ^s ⇔</font></div><div><font face="monospace, monospace"> is_set_theory mem ∧</font></div><div><font face="monospace, monospace"> is_inductive mem indset ∧</font></div><div><font face="monospace, monospace"> is_choice mem ch`</font></div></div></div><div><br></div><div>There is included a proof that the inductive axiom implies the prior infinity axiom; this proof depends on the Axiom of Regularity.</div><div><br></div><div><div><font face="monospace, monospace"> inductive_set_infinite</font></div><div><font face="monospace, monospace"> |- is_set_theory mem ∧ is_inductive mem indset ⇒</font></div><div><font face="monospace, monospace"> is_infinite mem indset</font></div></div><div><br></div><div>I also added several new set theory constants, such as binary intersection, inverse of a function, image of a function on a set, dependent function space, and dependent product space. The last three required the Axiom Schema of Replacement to define.</div><div><br></div><div>I am hoping to stimulate some discussion and debate. It is necessary to extend the existing partial specification of set theory to support the construction of the set-theory model of HOL-Omega. Besides that, there will probably be other uses for ZFC set theory, and I felt we needed a more complete axiomatic basis.</div><div><br></div><div>The next steps would presumably be the definition of the ordinals, with transfinite induction, and the cardinals.</div><div><br></div><div>Enjoy, and please let me know what you think.</div><div><br></div><div>Peter</div><div><div data-smartmail="gmail_signature"><div dir="ltr"><div><br></div><div><br></div>"In Your majesty ride prosperously <br>because of truth, humility, and righteousness;<br>and Your right hand shall teach You awesome things." (Psalm 45:4)</div></div>
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