Is 1 + 1 = 2 True?

Go up to Classic Fallacies index Go down to first subsection This is Not the Fallacy Go forward to 1=2: A Proof using Complex Numbers Switch to text-only version (no graphics) Go to University of Toronto Mathematics Network Home Page ... Step 1: Let a=b. ... Step 6: and . ... Step 8: and cancelling the from both sides gives 1=2. See if you can figure out in which step the fallacy lies. When you think you've figured it out, click on that step and the computer will tell you whether you are correct or not, and will give an additional explanation of why that step is or isn't valid.

1 plus 1 equals 2 because it represents the operation of addition where combining one unit with another unit yields two units, as per the principles of arithmetic. ... 1 + 1 = 21+1=2, we should recognize the fundamental principles of addition. Addition is an arithmetic operation where we combine ...

For example, they said stuff like ... apple plus one apple equals two apples" which you can't say in ZFC, and which is considered one of the basic flaws of PM (together with the axiom of reducibility, maybe), because it makes things ambiguous and harder to prove.ReplyDelete ... To prove 1+1=2, you have ...

1 * 1 = 2 for sufficiently large values of 1. ... In the english language, one plus one equals not just two but two ones, incomplete answers are not allowed . Eg test : A. 2 B. Apples C. 2 ones D. Zero C is the correct answer because it is complete

Also Cantor's proof that 2n > n breaks down unless n is finite." — 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:

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As you can see, it ends with "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." The theorem above, $\ast54\cdot43$, is already a couple of hundred pages into the book (Wikipedia says 370 or so)

The B.S. goes all the way back to the first two numbers you ever learned: 1 and 2. It turns out these seemingly separate digits are, in fact, exactly the same, and that 1 actually equals 2.

I've never studied the Principia in detail so cannot authoritatively claim that there isn't anything in this chapter that's relevant to the proof of the claim on page 362, but it does seem unrelated to me, and at least the first couple of pages of chapter III have rambling philosophical-sounding discussions about the meaning of statements such as "Socrates is mortal", "Scott is Scott", "Scott is the author of Waverley", etc, which clearly have nothing to do with the claim that 1+1=2. The claim that Russell and Whitehead "spent 362 pages to prove 1+1=2" is misleading in another way, since it suggests not only that the proof on page 362 relies in a logical sense on everything that precedes it (which as I said appears to be false), but also that proving this claim is the goal of all the preceding developments.