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Fundamental Knowledge-Part 2: Models

By Andrew on | Leave a comment

The next task is to absorb the traditional area of mathematical logic.  One key missing ingredient is a model.  Let us recall the traditional setup (taken from [1]).

Definition 2.1.  Let S be a set (of symbols).  An S-structure is a pair \mathfrak{A}=(A,\mathfrak{a}) where A is a nonempty set, called a universe, and \mathfrak{a} is a map sending symbols to elements, functions, and relations of A.  An assignment of an S-structure (A,\mathfrak{a}) is a map \beta:S\to A.  An S-interpretation is a pair \mathfrak{I}=(\mathfrak{A},\beta) where \mathfrak{A} is an S-structure and \beta is an assignment in \mathfrak{A}.

For shorthand notation, the convention (with some of my modifications) is to write:  c^\mathfrak{A}=\beta(c), (f(t_1,...,t_n))^\mathfrak{A}=\mathfrak{a}(f)(\beta(t_1),...,\beta(t_n)), and (xRy)^\mathfrak{A}=\beta(x)\mathfrak{a}(R)\beta(y).  These are the terms.  Formulas are then built from the terms using traditional (although this can be generalized) logical connectives.

The notion of a model is then defined via induction on formulas.

Definition 2.2.  Let \mathfrak{I}=(\mathfrak{A},\beta) be an S-interpretation.  We say that \mathfrak{I} satisfies a formula \phi (or is a model of \phi), denoted \mathfrak{I}\vDash\phi, if \phi^\mathfrak{A} holds, where \phi^\mathfrak{A} is defined via its components and \beta and \mathfrak{a} where necessary.

Formal languages in convention are built up from the formulas mentioned above, which are nothing more than special cases of Alt Definition 1.3.  A model for a language is hence nothing more than an A-interpretation into a structure, where A is an alphabet (provided it is equipped with a logic system).  This is precisely what I have constructed in Part 1;  the symbols of W\subset A^* are mapped to the universe \mathcal{L}_{F,T,W}.  The next thing to establish is that every model is a language model.  This is trivial since a model by definition satisfies a set of formulas as well as compounds of them (i.e. it must satisfy a language).  Hence we have no need to trouble ourselves with interpretations and may simply stick to the algebra of Part 1.

While we have absorbed model theory, there are a few more critical topics to absorb from mathematical logic. We return to the language of Part 1 (no pun).  Let X be a theory of \mathcal{L}_{F,T,W} and \varphi:F[X]\to V be a binary logic system.  A formula \phi\in\mathcal{L}_{F,T,W} is derivable in X if it is a proposition (i.e. is in F[X]).  We may write X\vdash\phi.  This definition is in complete agreement with the traditional definition (namely, there being a derivation, or finite number of steps, that begin with axioms and use inference rules);  it is nothing more than saying it is in F[X].  Similarly \phi\in\mathcal{L}_{F,T,W} is valid if \varnothing\vdash\phi, or equivalently, it is derivable in any theory.  In our setup this would imply \phi\in F[\varnothing]=\varnothing.  Hence no formula is valid.

Let F have a unary operation \lnot and \varphi:F[X]\to V be a logic system on a theory X.

If we assume \lnot to be idempotent (\lnot\lnot\phi=\phi), then since \varphi is a homomorphism, we have \varphi(\phi)=\varphi(\lnot\lnot\phi)=\lnot\lnot\varphi(\phi).  That is, the corresponding unary operation in V must also be idempotent on ran(\varphi).

Definition 2.3.  A unary operation \lnot (not necessarily idempotent) is consistent in \varphi if for all \phi\in F[X], \varphi(\phi)\neq\varphi(\lnot\phi).

If we assume \lnot is consistent in \varphi and that \varphi is a binary logic system, then the corresponding \lnot in V is idempotent since

\varphi(\phi)=0\Rightarrow\varphi(\lnot\phi)=1\Rightarrow\varphi(\lnot\lnot\phi)=\lnot\lnot\varphi(\phi)=0.

Again, proofs in a binary system are independent of the choice of valence.   If we assume consistency and idempotency, then we have a nonidentity negation which is idempotent on the range.  The case for assuming binary system and idempotency yields either a trivial mapping of propositions (all to 0 or all to 1), or that \lnot is consistent and idempotent on V.  And lastly if we assume all three (idempotency and consistency of \lnot together on a binary system), we obtain a surjective assignment with idempotent negation in V.

Let \varphi:F[X]\to V be a binary logic system where V is a boolean algebra.  Then the completeness and compactness theorems are trivial.  Recall these statements:

Completeness Theorem.  For all formulas \phi and models \mathfrak{I},

\mathfrak{I}\vDash\phi\Rightarrow X\vdash\phi

where \mathfrak{I}\vDash X.

Compactness Theorem.  For all formulas \phi and models \mathfrak{I},

X\vdash\phi\Rightarrow\mathfrak{I}\vDash\phi

where \mathfrak{I}\vDash X.

Traditionally these apply to, what we would call, a binary logic system \varphi:F[X]\to V where V is a boolean algebra (hence F has a consistent, idempotent negation) under traditional operations, and in particular this fixes the operational/relational structures of F, T, and W , but X is arbitrary.  In this setup, all “formulas” (or what we would hence call propositions since they are generated by a theory) are trivially satisfiable since they have a language model.  Hence Compactness is true.  Moreover since they are propositions in a binary logic system, they are in some F[X] for a theory X and are hence derivable; so we have Completeness.

Lastly we wish to address Godel’s Second Incompleteness Theorem;  recall its statement:

Godel’s Second Incompleteness Theorem.  A theory contains a statement of its own consistency if and only if it is inconsistent.

We have only defined what it means for a unary operation in a logical system to be consistent.  Hence we can say that a binary logic system with a unary operation is consistent if its unary operation is consistent.  But all of these traditional theorems of mathematical logic are assuming a binary logic system where V is a boolean algebra , \lnot is idempotent, and the map \varphi:F[X]\to V is surjective.  Hence \lnot is consistent (from above discussion), and the consequence in the theorem is false.

The weakest possible violation of the antecedent of Godel’s theorem is to use a structure to create itself (i.e. that it is self-swallowing), which makes no sense, let alone using it to create a larger structure within which is a statement about the initial structure.  That a binary logic system with unary operation could contain a statement of its own consistency is itself a contradiction, since the theory itself, together with the statement \phi, are in a metalanguage.  It is like saying that one need only the English language to describe the algebraic structure of the English language.  As we previously said at the end of Part 1, one can get arbitrarily close to doing this–using English to construct some degenerate form of English, but you can never have multiple instances of a single language in a language loop.  Another example would be having the class of all sets, then attempting to prove, using only the sets and operations of them, that there is a class containing them.

Hence the antecedent is also false.  So both implications are true.

[1]  Ebbinghaus, H.-D., J. Flum, and W. Thomas.  Mathematical Logic.  Second Edition.  Undergraduate Texts in Mathematics.  New York: Springer-Verlag.  1994.

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Fundamental Knowledge-Part 1: The Language Loop

By Andrew on | Leave a comment

We must start with a language.  A language can be defined in two ways.  First let us begin with the axioms of pairing, union, and powerset and schema of separation.  This gives us a cartesian product of sets, and hence functions.

Definition 1.1.  An n–ary operation on a set X is a map O:X^n\to X.  A structure is a set X together with an n-ary operation.  The signature of a structure X is the sequence (n_1,...,n_k,...) where n_k is the number of k-ary operations.

Definition 1.2.  Let X and Y be structures with the same signature such that each k-ary operation of X is assigned to a k-ary operation of Y (i.e. f(O_i)=O^i where O_i is the ith k-ary operation of X).  A homomorphism between structures X and Y is a map \varphi:X\to Y such that

\displaystyle \varphi(O_i(x_1,...,x_n))=O^i(\varphi(x_1),...,\varphi(x_n)).

Note that a nullary operation on X is a map O:\varnothing\to X.  That is, it is simply an element of X.  Now let A be a set which we will call an alphabet, and its elements will be called letters.  A monoid X has a nullary operation, 1\in X called a space, and a binary operation, which will simply be denoted by concatenation.  We define the free monoid on A as the monoid A^* consisting of all strings of elements in A.  We now have two definitions of a language, of which the first is traditional and the second is mine:

Definition 1.3.  A language is a subset of A^*.

Alt Definition 1.3.  Let W\subset A^*, T be a relational structure (a set together with an n-ary relation), and F be a structure.  The language \mathcal{L}_{F,T,W} is defined as F[T[W]] where X[Y] is the free X-structure on Y.  In particular elements of W are called words, elements of T[W] are called terms, and elements of \mathcal{L}_{F,T,W} are called formulas.

Definition 1.4.  A theory of \mathcal{L}_{F,T,W} is a subset X\subset\mathcal{L}_{F,T,W}.  Elements of a theory are called axioms.  Elements of F[X] are called propositions.  A theory X of \mathcal{L}_{F,T,W} is called a reduced theory if for all \phi,\psi\in X, \psi\neq O(\phi,x_1,...,x_{n-1}) for all n-ary operations of F and all placements of \phi in evaluation of the operation.  (That is, the theory is reduced if no axiom is in the orbit of another).

For example, the theory \mathcal{L}_{F,T,W} is called the trivial theory.  The theory \varnothing is called the empty (or agnostic) theory.

Definition 1.5.  An n–ary logic system on a theory X is a homomorphism \varphi:F[X]\to V where F and V have the same signature and V has cardinality n.  We may also say the logic system is normal if \varphi(\phi)=\varphi(\psi) for all \phi,\psi\in X.

In traditional logic V is a two element boolean algebra.  Traditional logic also has a special kind of function on its language.

Definition 1.6.  A quantifier on \mathcal{L}_{F,T,W} is a function \exists:T[W]\times\mathcal{L}_{F,T,W}\to\mathcal{L}_{F,T,W}.  We may write:

\exists(x\in X,\phi)=(\exists x\in X)\phi.

In particular it is a pseudo operation, and gives the language a pseudo structure.  This is similar to modules, where in this case a product of a term and a formula are sent to a formula.

Hence our initial assumption of four axioms (as well as the ability to understand the English language), have in turn given us the ability to create a notion of a language of which a degenerate English can be construed as a special case.  This is certainly circular in some sense, but in foundations we must appeal to some cyclic process.  One subtlety worth noting is that the secondary language created will always be “strictly bounded above” by the initial language;  they aren’t truly equivalent.  (In fact this last statement is similar to the antecedent of Godel’s Second Incompleteness theorem).

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A Crash Course on Mathematics’ Crash Course

By Andrew on | Leave a comment

There are two huge problems with the way math is taught at the secondary and lower-division level:

  1. It isn’t taught as what it is: language, logic, axioms, and consequences that form the foundation of knowledge and reason.
  2. The intense emphasis on algebraic operations of real numbers makes colloquial usage of the term “mathematics” an unrealized drastic overstatement.

Note that some mathematicians and logicians may refer to mathematics as a particular type of set theory and all of the theorems that follow; but I am more generally equating math to a formal language and all theorems that follow for any choice of a set of axioms and a logic system.

The key point not made (sometimes not even made in a mathematician’s career of coursework) is that math boils down to assumptions and consequences.  Philosophy (or what I call archaic mathematics [don’t be offended]) evolved into math once philosophers started making well-defined assumptions and proved what followed.  Prior to this, philosophers had assumed in fundamental yet immeasurable/unworkable objects like morality, God, existence, and truth.  Their mathematical descendants have respectively transformed these objects into utilitarianism, universe, arbitration, and valence on propositions—thereby allowing consistent consequences.  Now, centuries in the making, the ubiquity of the fundamentality (yet undefinability) of the former four objects expresses the collision course of mathematics:  if it crashes—if pseudoscience saturates convention, then logic will no longer be a mechanism capable of transporting people.

So, how do we solve the aforementioned problems?  One option is with a top-down approach.  If professors can solidify the more accurate interpretation of mathematics to students, then those students who go on to teach at the secondary and post-secondary levels can induct the process.  One may also attribute some of the blame to state legislation.  48 of the 50 states (the excluded being Texas and Alaska) have boarded the Common Core State Standards Initiative—mostly led by the National Governors Association and the Council of Chief State School Officers and consisting of “parents, teachers, school administrators and experts from across the country together with state leaders”.1  While I see usefulness in standard curriculum of mathematical operations, word problems, trig, and functions, I don’t see the need to spend 4+ years on it (assuming it’s taught efficiently).  I’m also not surprised that most students don’t see the use in it considering how far removed it is.  I think students could more easily learn concepts from logic and set theory.  Such concepts are foundational to math, and hence knowledge, and are correspondingly more applicable to everyday phenomena.  Moreover, a solid understanding of them can accelerate the students’ ability to learn the current standards.  Even if legislation permitted, it would also take a qualified teacher with a strong background in set theory and logic to make the connections.  This brings us to a funding issue, in particular that of salary incentives for qualified graduates.

Recall in the 1960s, after the launch of the Sputnik satellite by the Soviet Union, the US along with other European countries began a “New Math” campaign in which foundational math topics like logic and set theory were emphasized at an earlier age.  The program broke down within a decade due to pragmatic issues: parents not being able to help their kids (an illegitimate argument, since after all, it is a reform and reforms mean new things) and ineptitude of teachers unfamiliar with the concepts (a legitimate argument that could be tackled with incentives for qualified graduates, which could also mitigate the first issue).  The only other legitimate criticism I can find concerns the trade off of spending time on computational abilities versus investigative abilities.  While I find the first to be important, I claim high school and college graduates could benefit more from the latter.  I propose that primary education in math remain unchanged, and that we introduce logic, set theory, etc. at the high school level with problem sets that are roughly 50% computational and 50% proof based.

[1] http://www.corestandards.org/

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