Before writing Principles, Russell became aware of Cantor's proof that there was no greatest cardinal number, which Russell believed was mistaken. The Cantor Paradox in turn was shown (for example by Crossley) to be a special case of the Russell Paradox. This caused Russell to analyze classes, for it was known that given any number of elements, the number of classes they result in is greater than their number. This in turn led to the discovery of a very interesting class, namely, the class of all classes. It contains two kinds of classes: those classes that contain themselves, and those that do not. Consideration of this class led him to find a fatal flaw in the so-called principle of comprehension, which had been taken for granted by logicians of the time. He showed that it resulted in a contradiction, whereby Y is a member of Y, if and only if, Y is not a member of Y. This has become known as Russell's paradox, the solution to which he outlined in an appendix to Principles, and which he later developed into a complete theory, the Theory of types. Aside from exposing a major inconsistency in naive set theory, Russell's work led directly to the creation of modern axiomatic set theory.

Russell continued to defend logicism, the view that mathematics is in some important sense reducible to logic, and along with his former teacher, Alfred North Whitehead, wrote the monumental Principia Mathematica, an axiomatic system on which all of mathematics can be built. The first volume of the Principia was published in 1910, and is largely ascribed to Russell. More than any other single work, it established the specialty of mathematical or symbolic logic. Two more volumes were published, but their original plan to incorporate geometry in a fourth volume was never realized, and Russell never felt up to improving the original works, though he referenced new developments and problems in his preface to the second edition. Upon completing the Principia, three volumes of extraordinarily abstract and complex reasoning, Russell was exhausted, and he felt his intellectual faculties never fully recovered from the effort.[10] Although the Principia did not fall prey to the paradoxes in Frege's approach, it was later proven by Kurt G?del that neither Principia Mathematica, nor any other consistent system of primitive recursive arithmetic, could, within that system, determine that every proposition that could be formulated within that system was decidable, i.e. could decide whether that proposition or its negation was provable within the system

This isn't the Godel thread, it's the 2+2 thread. But parenthetically, you MIGHT be able to find a trained mathematician or logician who could make the case that since Godel it is not a foregone conclusion that 2+2=4 or whatever. I don't know. I'm willing to accept what biographers and synthesizers of his work will call his implications. I'm not going to pretend that I can follow his mathematics and translate it to the 2+2=4 question, but I can tell you that Bertrand Russell, who is regarded along with Godel as one of the two leading logicians in the 20th century, spent a good chunk of his career trying to turn simple arithmetic into statements of pure logic and gave up / failed after Godel. That's pretty striking. If he'll give up that basic arithmetic effort in the face of Godel, then what does that mean for calculus?

... 2+2 = 2+2 ... or, anything else by other standards...:a-ok:

People often use terms loosely. Suppose I were to say I think you are an idiot. That would be using the word, "idiot" loosely. But if I said that someone with an I.Q. of 50 was an idiot, I would be using the term "idiot" strictly.

But it was not because of Godel that he gave up. He gave up because Frege showd a contradiction in his definition of "number". The class paradox.

If you look at the last sentence of the Wiki quote in your post, you will see that it confirms what I said about Godel.

Originally Posted by Aedes
This isn't the Godel thread, it's the 2+2 thread. But parenthetically, you MIGHT be able to find a trained mathematician or logician who could make the case that since Godel it is not a foregone conclusion that 2+2=4 or whatever. I don't know. I'm willing to accept what biographers and synthesizers of his work will call his implications. I'm not going to pretend that I can follow his mathematics and translate it to the 2+2=4 question, but I can tell you that Bertrand Russell, who is regarded along with Godel as one of the two leading logicians in the 20th century, spent a good chunk of his career trying to turn simple arithmetic into statements of pure logic and gave up / failed after Godel. That's pretty striking. If he'll give up that basic arithmetic effort in the face of Godel, then what does that mean for calculus?



ken,
Where did Frege show a contradiction in his definition of "number".
"The class paradox"?? Do you mean Russell's Paradox?

Why did you ask when you already knew? Russell's paradox is also called "the class paradox".

Ken: But it was not because of Godel that he gave up. He gave up because Frege showd a contradiction in his definition of "number". The class paradox.

It was Russell who showed a contrradiction in Frege, not as you claim.

PM, as it is often abbreviated (not to be confused with Russell's 1903 Principles of Mathematics), is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic. One of the main inspirations and motivations for PM was Frege's earlier work on logic, which had led to paradoxes discovered by Russell. These were avoided in PM by building an elaborate system of types: a set of elements is of a different type than is each of its elements (set is not the element; one element is not the set) and one cannot speak of the "set of all sets" and similar constructs, which lead to paradoxes (see Russell's paradox).


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.)

In 1930, G?del's completeness theorem showed that propositional 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 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.
G?del's incompleteness theorems cast unexpected light on these two related questions.

G?del's first incompleteness theorem showed that Principia could not be both consistent and complete. According to the theorem, for every sufficiently powerful 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: 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.

G?del's second incompleteness theorem 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).

He gave up PM chiefly because of Godel, but also because (it seems) he got burnt out with it and it was no longer profitable to publish. Frege inspired it -- to say he took it down is, frankly, wrong.

Principia Mathematica - Wikipedia, the free encyclopedia

11 (in base 3)....I hope I did that right. Who uses base3?

(1 + 1) + (1+ 1) = 1 + 1 + 1 + 1

The point is it's all one number, stuck together with the same glue.

11 (in base 3)....I hope I did that right. Who uses base3?

Not when squared.

The Mayans used base 4.

Not when squared.

I admit it gets complex when we deal with negative numbers, and roots. But as long as a number is rational, it can be broken into ones, yes? And can't all rational numbers be written in base 2/binary?

I don't know about all rational numbers.

e.g. 0.5 or 1/2

1 divided by (1+1) = half of 1

or something like that.