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In different words, no false statement can be proved within the system, and, perhaps most importantly

as far as Godel is concerned,  ". . . there are no statements which the system regards as both true and

false."9 These two, consistency and completeness, are necessary for an exhaustive, contradiction free,

formal system.  And the claim made by Russell and Whitehead was that their formal system, the

Principia, fit the bill.  They had no proof, however   you could not really prove a work like the Principia

  and in 1931 Godel's famous paper was published; it proved that the Principia, and any other similar

system that could be developed, would never fulfill both consistency and completeness simultaneously.

One or the other, yes; but not the two together.  In fact, Godel's method demonstrated that the cause of

this is actually the system's inherent strength   the "system's own richness brings about its own down 

fall."10 His theorem is stunningly correct.

Godel looked at Principia Mathematica, a system that was supposed to produce all the truths

involving natural number theory.  He then demonstrated that, under the rules of the Russell Whitehead

system, he could legitimately construct a true self referential statement, P, that asserts: "This statement

cannot be proven true by the system outlined in Principia Mathematica."  He also showed that, in the

system, there is a complex, polynomial equation that is true only when P is true.  Now, note that

Principia Mathematica is deemed sufficiently powerful to be consistent; therefore, it can only generate

true theorems.  So, then, the theorem is true; but that means that it cannot be proven within the sys 

tem!  Godel showed, then, that it could neither be proven nor disproven   it is an "undecidable propo 

sition."  So, in fact, the very consistency of the system succeeds in bringing about its own incomplete 

ness.  Extrapolating, "any proof of consistency of a logical system . . . implies inconsistency in that sys 

tem."11 And if the statement in question is false, well, then the system has generated a false proposi 

tion   it is radically inconsistent.  Godel also showed that even if one were to add this statement as yet

another axiom (so as to form a new system, thus not having to worry about proving the statement), there

would still be another contradiction causing statement in this new system.  One, of course, could also

add this new statement as an axiom to get another new system, but no matter how many times this were

done, there would still be more statements just like it   logically independent of the axioms, statements

that the axioms in each respective system were not sufficiently strong either to establish or to refute.

Hofstadter calls this "essential incompleteness,"12 because it is part and parcel of all the systems.  A sys 

tem is thus imperfect how it is.  Make it complete and you prove inconsistency stronger, because you

can generate a true statement that proves inconsistency; make it weaker and your system either gen 

erates false statements or it is incapable of generating all the needed true statements (and is incom 

plete).  And, thus, the hopes of Russell, Whitehead, Hilbert and others were shattered.  Godel had shown

that ". . . there is an endless number of true arithmetical statements which cannot be formally deduced

from any given set of axioms by a closed set of rules of inference.  It follows that an axiomatic approach

to number theory cannot exhaust the domain of arithmetical truth."13 There is a basic limit, according

to Godel, of what can be done with a formal system; and the only way around that limit is through incon 

sistency, or by reasoning outside of the system, which defeats the original purpose, the formalization of

a contradiction free, closed system.  By going outside of the system, you now have a new system, one

that can only be taken care of again by going outside of it, and so on ad infinitum.  So mathematics, and

consistent logical systems in general, remain incomplete.                                                                  

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Elements of Godel's Incompleteness in Axiomatic Systems      Jared Balkman