Why are reals uncountable

Explain which real numbers have more than one decimal expansion, and then explain why the real number constructed in the proof is guaranteed not to be on the list of real numbers. Home » Functions » Uncountability of the Reals. Collapse menu 1 Logic 1. Logical Operations 2. Quantifiers 3. De Morgan's Laws 4. Mixed Quantifiers 5. Logic and Sets 6. Families of Sets 2 Proofs 1.

Direct Proofs 2. Divisibility 3. Existence proofs 4. Induction 5. Uniqueness Arguments 6. Indirect Proof 3 Number Theory 1. Congruence 2. The Euclidean Algorithm 4. The Fundamental Theorem of Arithmetic 6.

The Chinese Remainder Theorem 8. The Euler Phi Function 9. The Phi Function—Continued Wilson's Theorem and Euler's Theorem Public Key Cryptography Quadratic Reciprocity 4 Functions 1. Definition and Examples 2. Induced Set Functions 3. Injections and Surjections 4. More Properties of Injections and Surjections 5. Pseudo-Inverses 6. Bijections and Inverse Functions 7. It's not the first number because it's different in the first place, it's not the second number because it's different in the second place, it's not the third number because it's different in the third place, it's not the fourth number because it's different in the fourth place, etc.

Thus, by contradiction, the interval is not countable. Well, suppose there isn't - that Cantor's conclusion, his theorem , is wrong, because our enumeration covers all real numbers.

Wonderful, but let us see what happens when we take our enumeration and apply Cantor's diagonal technique to obtain a real number that can't be in this sequence. But that contradicts our supposition! Hence our supposition - that we can have an enumeration of all reals - is false. That the argument can be applied to any enumeration is what it takes for Cantor's theorem to be true. Section 7, dealing with how the counterfactual assumption confuses, might be of particular interest.

Cantor's diagonal is a trick to show that given any list of reals, a real can be found that is not in the list. Cantor's diagonal is a clever solution to finding a number which satisfies these properties. The number which is the diagonal is transformed s. Thus the number is unique to the list. Sign up to join this community. The best answers are voted up and rise to the top.

Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why is the set of all real numbers uncountable? Ask Question. Asked 9 years ago. Active 2 years, 10 months ago. Viewed 16k times. Lets say I assign the following numbers ad infinitum Mirrana Mirrana 8, 33 33 gold badges 77 77 silver badges bronze badges.

Cantor explicitly tells us what that "one" real number is Presumably, if he knew what the Lebesgue measure was, he would already be comfortable with Cantor's argument. Show 7 more comments. Active Oldest Votes. It just shows that there are no less real numbers then natural ones. But I guess my issue is with the fact that I don't understand this well enough yet. I only just learned about countability, and even then very poorly, since my prof is not a good teacher.

Add a comment. Let's say the list goes like: 0. Check out this YouTube clip. Fly by Night Fly by Night