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|Description: (Theorems equidK 28136 through ax12dgen4K 28181 are part of a study of our
non-Tarski predicate calculus axiom schemes. We are using this theorem
as a placeholder to describe this study.)
The orginal axiom schemes of Tarski's predicate calculus are ax-5 1533, ax-8 1623, ax-9v 1632, ax-13 1625, ax-14 1626, and ax-17 1628 (see http://us.metamath.org/mpegif/mmset.html#compare) and are shown as axiom schemes B4 through B8 in [KalishMontague] p. 81. These are shown to be logically complete by Theorem 1 of [KalishMontague] p. 85.
The axiom system of set.mm includes the additional axiom schemes ax-6 1534, ax-7 1535, ax-11 1624, and ax-12 1633, which are not part of Tarski's axiom schemes. They are used (and we conjecture are required) to make our system "metalogically complete" i.e. able to prove directly all possible schemes with wff and set metavariables, bundled or not, whose object-language instances are valid. (ax-11 1624 has been proved to be required; see http://us.metamath.org/award2003.html#9a. Metalogical independence of the other three are open problems.)
(There are additional predicate calculus axiom schemes included in set.mm such as ax-4 1692, but they can all be proved as theorems from the above.)
Terminology: Two set (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the and in ax-9 1684 are bundled, but they are not in ax-9v 1632. We also say that a scheme is bundled when it has at least one pair of bundled set metavariables. If distinct variable conditions are added to all set metavariable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax-9v 1632 is the principal instance of ax-9 1684. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance of ax-9 1684 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($d) to be concerned with. There is also a small economy in being able to state principal and degenerate instances simultaneously. A disadvantage is that bundling may present difficulties in translations to other proof languages, which typically lack the concept (in part because their variables often represent the variables of the object language rather than metavariables ranging over them).
Because Tarski's axiom schemes are logically complete, they can be used to prove any object-language instance of ax-6 1534, ax-7 1535, ax-11 1624, and ax-12 1633. "Translating" this to Metamath, it means that Tarksi's axioms can prove any substitution instance of ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633 in which (1) there are no wff metavariables and (2) all set metavariables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each set metavariable is an object-language variable. (There may also be specific instances with wff metavariables and/or bundling that are directly provable from Tarski's axiom schemes, but it isn't guaranteed. Whether all of them are possible is part of the still open metalogical independence problem for our additional axiom schemes.)
It can be useful to see how this can be done, both to show that our additional schemes are valid metatheorems of Tarski's system and to be able to translate object language instances of our proofs into proofs that would work with a system using only Tarski's original schemes. In addition, it may (or may not) provide insight into the conjectured metalogical independence of our additional schemes.
Past work showed that instances of ax-11o 1941 meeting condition (1) can be proved without invoking that axiom scheme (see comments in ax-11 1624). However, it was somewhat awkward to use, involving an inductive argument with auxiliary theorems ax11eq 2108, ax11el 2109, ax11indn 2111, ax11indi 2112, and ax11inda 2116. It also used axiom schemes other than Tarski's.
The new theorem schemes ax6wK 28166, ax7wK 28169, ax11wK 28174, and ax12wK 28177 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-6 1534, ax-7 1535, ax-11 1624, and ax-12 1633 meeting conditions (1) and (2). (The "K" suffix stands for Kalish/Montague, whose paper was a source for some of the proofs. I may change these names in the future since our practice has been to reserve upper case for special cases such as the ALT or OLD suffixes.) Each hypothesis of ax6wK 28166, ax7wK 28169, and ax11wK 28174 is of the form where is an auxiliary or "dummy" wff metavariable in which doesn't occur. We can show by induction on formula length that the hypotheses can be eliminated in all cases meeting conditions (1) and (2). The example ax11wdemoK 28176 illustrates the techniques (equality theorems and bound variable renaming) used to achieve this.
We also show the degenerate instances for axioms with bundled variables in ax7dgenK 28171, ax11dgenK 28175, ax12dgen1K 28178, ax12dgen2K 28179, ax12dgen3K 28180, and ax12dgen4K 28181. (Their proofs are trivial but we include them to be thorough.) Combining the principal and degenerate cases outside of Metamath, we show that the bundled schemes ax-6 1534, ax-7 1535, ax-11 1624, and ax-12 1633 are schemes of Tarski's system, meaning that all object language instances they generate are theorems of Tarski's system.
It is interesting that Tarski's system bundles set metavariables in ax-8 1623, ax-13 1625, and ax-14 1626; indeed, a degenerate instance of ax-8 1623 appears to be indispensable for the proof of equidK 28136. Perhaps his general philosophy was that bundling is acceptable for free variables. But he also used the bundled scheme ax-9 1684 in an older system, so it seems the main purpose of his later ax-9v 1632 was just to show that the weaker unbundled form is sufficient rather than an aesthetic objection to bundled free and bound variables.
The case of ax-4 1692 is curious: originally an axiom of Tarski's system, it was proved redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the compact scheme form apparently cannot be proved directly from Tarski's other axioms. The best we can do seems to be ax4wK 28153, again requiring substitution instances of that meet conditions (1) and (2) above. Note that our direct proof ax4 1691 requires ax-11 1624, which is not part of Tarski's system.
(End of study description.)
|1||ax-9v 1632||. . 3|
|2||ax-8 1623||. . . . . 6|
|3||2||pm2.43i 45||. . . . 5|
|4||3||con3i 129||. . . 4|
|5||4||alimi 1546||. . 3|
|6||1, 5||mto 169||. 2|
|7||ax-17 1628||. 2|
|8||6, 7||mt3 173||1|
|Colors of variables: wff set class|
|Syntax hints: wn 5 wal 1532|
|This theorem is referenced by: equcomiK 28137 ax9dgenK 28158 ax12dgen1K 28178 ax12dgen3K 28180 ax12dgen4K 28181|
|This theorem was proved from axioms: ax-1 7 ax-2 8 ax-3 9 ax-mp 10 ax-5 1533 ax-gen 1536 ax-8 1623 ax-17 1628 ax-9v 1632|
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