Editing Principia Mathematica (section)

== Consistency and criticisms ==

According to [[Carnap]]'s "Logicist Foundations of Mathematics", Russell wanted a theory that could plausibly be said to derive all of mathematics from purely logical axioms. However, ''Principia Mathematica'' required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the [[axiom of infinity]], the [[axiom of choice]], and the [[axiom of reducibility]]. Since the first two were existential axioms, Russell phrased mathematical statements depending on them as conditionals. But reducibility was required to be sure that the formal statements even properly express statements of real analysis, so that statements depending on it could not be reformulated as conditionals. [[Frank Ramsey (mathematician)|Frank Ramsey]] tried to argue that Russell's ramification of the theory of types was unnecessary, so that reducibility could be removed, but these arguments seemed inconclusive.

Beyond the status of the axioms as [[logical truth]]s, one can ask the following questions about any system such as PM:
* whether a contradiction could be derived from the axioms (the question of [[inconsistency]]), and
* whether there exists a [[mathematical statement]] which could neither be proven nor disproven in the system (the question of [[completeness (logic)|completeness]]).

[[Propositional logic]] itself was known to be consistent, but the same had not been established for ''Principia'''s axioms of set theory. (See [[Hilbert's second problem]].) Russell and Whitehead suspected that the system in PM is incomplete: for example, they pointed out that it does not seem powerful enough to show that the cardinal ℵ<sub>ω</sub> exists. However, one can ask if some recursively axiomatizable extension of it is complete and consistent.

===Gödel 1930, 1931===

In 1930, [[Gödel's completeness theorem]] showed that first-order predicate logic itself was complete in a much weaker sense—that is, any sentence that is unprovable from a given set of axioms must actually be false in some [[Model theory|model]] of the axioms. However, this is not the stronger sense of completeness desired for ''Principia Mathematica'', since a given system of axioms (such as those of ''Principia Mathematica'') may have many models, in some of which a given statement is true and in others of which that statement is false, so that the statement is left undecided by the axioms.

[[Gödel's incompleteness theorems]] cast unexpected light on these two related questions.

Gödel's first incompleteness theorem showed that no recursive extension of ''Principia'' could be both consistent and complete for arithmetic statements. (As mentioned above, Principia itself was already known to be incomplete for some non-arithmetic statements.) According to the theorem, within every sufficiently powerful recursive [[logical system]] (such as ''Principia''), there exists a statement ''G'' that essentially reads, "The statement ''G'' cannot be proved." Such a statement is a sort of [[Catch-22 (logic)|Catch-22]]: if ''G'' is provable, then it is false, and the system is therefore inconsistent; and if ''G'' is not provable, then it is true, and the system is therefore incomplete.

[[Gödel's second incompleteness theorem]] (1931) shows that no [[formal system]] extending basic arithmetic can be used to prove its own consistency. Thus, the statement "there are no contradictions in the ''Principia'' system" cannot be proven in the ''Principia'' system unless there ''are'' contradictions in the system (in which case it can be proven both true and false).

=== Wittgenstein 1919, 1939 ===
By the second edition of ''PM'', Russell had removed his ''axiom of reducibility'' to a new axiom (although he does not state it as such). Gödel 1944:126 describes it this way:
{{Cquote|This change is connected with the new axiom that functions can occur in propositions only "through their values", i.e., extensionally (...) [this is] quite unobjectionable even from the constructive standpoint (...) provided that quantifiers are always restricted to definite orders". This change from a quasi-''intensional'' stance to a fully ''extensional'' stance also restricts [[predicate logic]] to the second order, i.e. functions of functions: "We can decide that mathematics is to confine itself to functions of functions which obey the above assumption". |source=''PM'' 2nd edition p.&nbsp;401, Appendix C}}

This new proposal resulted in a dire outcome. An "extensional stance" and restriction to a second-order predicate logic means that a propositional function extended to all individuals such as "All 'x' are blue" now has to list all of the 'x' that satisfy (are true in) the proposition, listing them in a possibly infinite conjunction: e.g. ''x<sub>1</sub>'' ∧ ''x<sub>2</sub>'' ∧ . . . ∧ ''x<sub>n</sub>'' ∧ . . .. Ironically, this change came about as the result of criticism from [[Ludwig Wittgenstein]] in his 1919 ''[[Tractatus Logico-Philosophicus]]''. As described by Russell in the Introduction to the Second Edition of ''PM'':
{{Cquote|There is another course, recommended by Wittgenstein† (†''Tractatus Logico-Philosophicus'', *5.54ff) for philosophical reasons. This is to assume that functions of propositions are always truth-functions, and that a function can only occur in a proposition through its values. (...) [Working through the consequences] it appears that everything in Vol. I remains true (though often new proofs are required); the theory of inductive cardinals and ordinals survives; but it seems that the theory of infinite Dedekindian and well-ordered series largely collapses, so that irrationals, and real numbers generally, can no longer be adequately dealt with. Also Cantor's proof that 2<sup>n</sup> > ''n'' breaks down unless ''n'' is finite." |source=''PM'' 2nd edition reprinted 1962:xiv, also cf. new Appendix C)}}
In other words, the fact that an infinite list cannot realistically be specified means that the concept of "number" in the infinite sense (i.e. the continuum) cannot be described by the new theory proposed in ''PM Second Edition''.

Wittgenstein in his ''Lectures on the Foundations of Mathematics, Cambridge 1939'' criticised ''Principia'' on various grounds, such as:
* It purports to reveal the fundamental basis for arithmetic. However, it is our everyday arithmetical practices such as counting which are fundamental; for if a persistent discrepancy arose between counting and ''Principia'', this would be treated as evidence of an error in ''Principia'' (e.g., that Principia did not characterise numbers or addition correctly), not as evidence of an error in everyday counting.
* The calculating methods in ''Principia'' can only be used in practice with very small numbers. To calculate using large numbers (e.g., billions), the formulae would become too long, and some short-cut method would have to be used, which would no doubt rely on everyday techniques such as counting (or else on non-fundamental and hence questionable methods such as induction). So again ''Principia'' depends on everyday techniques, not vice versa.

Wittgenstein did, however, concede that ''Principia'' may nonetheless make some aspects of everyday arithmetic clearer.

===Gödel 1944===
[[Kurt Gödel|Gödel]] offered a "critical but sympathetic discussion of the logicistic order of ideas" in his 1944 article "Russell's Mathematical Logic".{{sfn|Kleene|1952|p=46}} He wrote:
{{blockquote|It is to be regretted that this first comprehensive and thorough-going presentation of a mathematical logic and the derivation of mathematics from it [is] so greatly lacking in formal precision in the foundations (contained in '''✱1–✱21''' of ''Principia'' [i.e., sections '''✱1–✱5''' (propositional logic), '''✱8–14''' (predicate logic with identity/equality), '''✱20''' (introduction to set theory), and '''✱21''' (introduction to relations theory)]) that it represents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism. Syntactical considerations are omitted even in cases where they are necessary for the cogency of the proofs ... The matter is especially doubtful for the rule of substitution and of replacing defined symbols by their ''definiens'' ... it is chiefly the rule of substitution which would have to be proved.<ref name="Gödel 1944">{{harvnb|Gödel|1944|p=126}} (reprinted in {{harvnb|Gödel|1990|p=120}}).</ref>}}