User:Lord Farin/Sandbox

This page exists for me to be able to test features I am developing. Also, incomplete proofs may appear here.

Feel free to comment.

Over time, stuff may move to User:Lord_Farin/Sandbox/Archive.

Subpages of this one may exist; they are listed at this PW special page.

= Reworking of Definition:Natural Numbers =

Will separate Definition:Natural Numbers into axiomatisations and constructions. Will link these with appropriate theorems. Goal is to obtain proper modular structure.

Upon success, project will be repeated for the other number sets.

See User:Lord_Farin/Sandbox/Definition:Natural Numbers for a draft.

Prior to setting up the proof structure accompanying this, it will be required to rework the existing approaches so as to be fully compliant with recent house style. This in order to prevent duplicate work and make the transition easier. Such applies to all four approaches cited on the draft page.

The following approaches are present on :

Axiomatic:
 * Axiom:Peano's Axioms
 * Definition:Naturally Ordered Semigroup

Constructive:
 * Definition:Minimal Infinite Successor Set / Natural Numbers as Successor Sets
 * Natural Numbers as Cardinals

1-based Axiomatic:
 * Axiom:Axiom Schema for 1-Based Natural Numbers

1-based Constructive:
 * Natural Numbers in Real Numbers

Will need to contemplate how to cover the multiple versions of PA. Any sandboxed pages will need to be reviewed to comply with these changes (links, axiom numbering, extra proofs).



A "+" in front means the page is being worked on.


 * Definition:Axiomatic Definition of Natural Numbers
 * +Definition:One


 * Axiom Schemas for Natural Numbers
 * Identity Element of Natural Number Multiplication is One
 * Principle of Mathematical Induction
 * Principle of Finite Induction

Peano

 * Definition:Peano Structure
 * Definition:Addition
 * Definition:Successor Mapping
 * Definition:Successor
 * Definition:Successor Mapping on Natural Numbers


 * 1+1 = 2
 * Equivalence of Peano Axiom Schemas
 * Minimal Infinite Successor Set Fulfils Peano Axioms
 * Natural Numbers form Naturally Ordered Semigroup
 * Naturally Ordered Semigroup Satisfies Peano's Axioms
 * +Non-Successor Element of Peano Axiom Schema is Unique
 * Peano's Axioms Uniquely Define Natural Numbers
 * +Peano Structure Without Non-Successor Element
 * Principle of Finite Induction/Peano's Axioms
 * +Successor Mapping in Peano Set has No Fixed Element

Naturally Ordered Semigroup

 * +Definition:Zero of Naturally Ordered Semigroup


 * Natural Numbers form Naturally Ordered Semigroup
 * Naturally Ordered Semigroup Exists
 * Naturally Ordered Semigroup Satisfies Peano's Axioms
 * Naturally Ordered Semigroup Unique up to Isomorphism
 * +Smallest Element of Zero Complement of Naturally Ordered Semigroup
 * +Zero Complement
 * +Zero is Identity in Naturally Ordered Semigroup

Minimal Infinite Successor Set

 * Minimal Infinite Successor Set Fulfils Peano Axioms

Work List


This is in progress. The Definitions/Formal Systems category has been reworked for the directly relevant part. (I've let Bourbaki and some pages on Definition:String resp. Definition:Word (Formal Systems) rest for now.)

I will have to do a lot of reading in the Definitions/Logic category and its subcategories to try and bring order in the fragmentation. &mdash; Lord_Farin (talk) 12:27, 8 September 2013 (UTC)

User:Lord_Farin/Sandbox/LogicCategories will contain an attempt to classify and categorise the pages. Hopefully, it will largely overlap with the existing system. &mdash; Lord_Farin (talk) 12:24, 9 September 2013 (UTC)

The below two radical proposals seem to clear major problems with categorisation and clarity of the site. &mdash; Lord_Farin (talk) 18:09, 9 September 2013 (UTC)

 Ditch the Category:Mathematical Logic and Category:Definitions/Mathematical Logic categories. For, most, if not all, of the material currently there is one of:


 * 1) Actually more general, and applicable to either the Symbolic Logic or the Formal Systems categories;
 * 2) Part of computability/recursion theory;
 * 3) Applicable only to predicate logic/model theory

Now, although we have at least historical and sourced merit for pages like Definition:Mathematical Logic, the grand scheme of universal coverage that we strive for on will cause us to move material presented by authors in the context of mathematical logic according to one of the three rules (particularly the first and last). We thus do not need to cripple ourselves and artificially fragment the site by forcing this mould onto our system. Please post your thoughts. &mdash; Lord_Farin (talk) 15:03, 9 September 2013 (UTC)


 * I understand "mathematical logic" as being a more specific category than symbolic logic but more general than formal systems. It is the general framework into which formal systems goes, and requires as its starting-point a working model of number theory. It is the category in which Godel's Theorem lies. SO I would say: keep it, make Formal Systems a subcategory of it, and make it a subcategory of Logic (or Symbolic Logic, or both), but it definitely needs to be in there as there is a wealth of literature that refers to "mathematical logic".


 * It may turn out, on revisiting everything in that category in turn, that everything belongs in "formal systems" or "model theory" or whatever other categories we identify as a subcategory of "mathematical logic", but this will not invalidate the existence of the category itself. --prime mover (talk) 18:34, 9 September 2013 (UTC)


 * I read formal systems as being more or less the most basic category. I think it could be applied to defining programming languages, e.g. letting the theorems correspond to properly compiling programs.


 * But since the precise (or perhaps it's better to say "intuitive") definitions of, and interplay between, "mathematical logic", "symbolic logic", and "formal system" are not universal, it's at least clear that we need to do something to separate them (if appropriate). I'll get cracking on the other radical proposal first, then conduct a literature search. &mdash; Lord_Farin (talk) 21:41, 9 September 2013 (UTC)

 Merge Category:Propositional Logic and Category:Propositional Calculus, similarly for Category:Predicate Logic and Category:Predicate Calculus.

I have not been able to find an explanation for any distinction between these terms. They also seem to be used interchangeably, both on and in sources. At least, their distinction is not clear to the reader, and I deem it not useful. We might want to rename Definition:Propositional Calculus to Definition:Language of Propositional Logic or something like that. Please also post your thoughts on this one. &mdash; Lord_Farin (talk) 18:09, 9 September 2013 (UTC)


 * I think I'm inclined to agree. "Language of Propositional Logic" is a propositional calculus, and the latter term can remain in, but, instead of a category, as an appropriately written definition page.


 * Bear in mind that some of this material dates from the very early days of and we hadn't got to grips with how versatile we could structure it, so we (or I did, anyway) were constrained by what we thought were limitations of the medium. So I'm happy for that to happen. --prime mover (talk) 18:34, 9 September 2013 (UTC)

In PropLog
This section is about long-term and postponed things. For the short-term part, see User:Lord_Farin/Sandbox/PropLog.

Literature suggests that what is currently known as propositional tableaus would be more accurately described as analytic tableaus, contrasting the newly written semantic tableaus.

In due time, we will need to set up the following pages:

This will ensure complete rigour, but it's also quite... boring, to say the least. Alas, I don't see a way around it. It's kind of a basic version of what is known as "definitional equivalence" or "interdefinability" in proper model theory (where we can rely safely on a unique Prop/PredLog base): that two theories are strong enough to define each others' operations, relations, and constants.
 * Definition:Translation Scheme for Language of Propositional Logic, in which back-and-forth techniques to transliterate the wffs of respective formal languages into "our" FL, and back.
 * This includes noting that Huth-Ryan call the "Labeled Tree WFF" associated to a "normal WFF" a parse tree. This can be used to replace the source entry for that book on Definition:Language of Propositional Logic/Labeled Tree.
 * Definitions for any type of semantics set out in a source using a non-official FL.
 * Equivalence proofs to "intertwine" the semantics in the non-official FL and the transliteration.

Prolonging my thought, there may be a call for similar things in any proof-theoretical context. The nice thing is that this approach, once done, gives complete, formal (as opposed to intuitive) justification for using the PW formalisation of Prop/PredLog everywhere. Now it remains to be motivated for this massive undertaking... &mdash; Lord_Farin (talk) 19:11, 11 September 2013 (UTC)

At some point, the unique parsability of the various PropLog languages will need to be established. This will probably need some things like WFF of PropCalc is Balanced. &mdash; Lord_Farin (talk) 13:57, 16 September 2013 (UTC)

Other (mostly small) things:


 * Determine what to do with Unique Truth Value for WFF of PropLog under Model -- it's obsoleted by the reference to Principle of Definition by Structural Induction on Definition:Boolean Interpretation. Perhaps this specific result can remain as Truth Value of WFF under Boolean Interpretation is Well-Defined -- in due course, we should be able to also have Propositional Logic firmly grounded in the "length of WFF" induction type. But essentially, the proof currently up is not but an instane of said Principle of Definition by Structural Induction.
 * Thorough inspection of correctness of links in the logic categories -- often, it's links to symbolic defs when non-symbolic is intended and vice versa. Huge entangled mess. Progress to be recorded on User:Lord_Farin/Sandbox/PropLog.
 * Disambiguate Definition:Truth Value and Definition:Boolean Interpretation/Truth Value.
 * Uniformise notation for WFFs of PropLog. We now have $\mathbf A$, $P$, $\phi$, $p$, and others. Suggestions?
 * It seems nice to reserve $p,q,r$ for letters. Capitals seem largely extinct and are prone to confusion with other concepts. Tradition favours $\phi$ etc., but perhaps $\mathbf A$ etc. are better. I lean towards the Greek letters, at least for PredLog. It's complicated, since the Greek letters also often stand for mappings. &mdash; Lord_Farin (talk) 13:59, 7 December 2013 (UTC)
 * Since you ask ... My personal preference would be $\mathbf A$ etc. This is what's used by, for example, Keisler and Robbin, which (for all of its perceived faults) is elegant in presentation. --~


 * To separate results about strings/words from the Formal Systems category.
 * To graft the Bourbaki pages as just another branch onto the Formal Language/Predicate Calculus definitions tree.
 * To separate any symbolic treatment of logic (such as present in Definition:Converse) from the Definitions/Logic category.
 * To move computability theory stuff out of the Category:Definitions/Mathematical Logic category as it doesn't belong there.
 * Disambiguate the links to Definition:Axiom.


 * Category:Definitions/Logical Words for the array of logical words: words in natural language that have specified mathematical-logical meaning, particularly when used in proofs.