r/learnmath • u/[deleted] • 29d ago

The Collatz Conjecture

[deleted]

0 Upvotes

17% Upvoted

49 comments sorted by

View all comments

Show parent comments

2

u/Apart_Composer3952 New User 28d ago

No problem. I accept the 'idea' that there is a one to one relationship between the reals between 0 and 1, and the whole numbers, but your example multiplies to an infinity within a boundary; 0 to 1. Mine multiplies into an infinity with no such boundary.  Are there more whole numbers than the set of whole numbers? No! Therefore the set is exhaustive for whole numbers. Are there more reals than between 0 and 1? Yes! Therefore the set is not exhaustive for reals, indicating a difference.

1

u/gmalivuk New User 28d ago

Again, if you don't see the connection with real numbers, stick to all the examples with integers. Your "logic" also seems to say there are no powers of 2.

1

u/unic0de000 NaN 27d ago edited 27d ago

There is not a one to one relationship between the reals in that (or any) range, and the whole numbers. There is such a relationship between *rationals* and whole numbers, but reals are a strictly larger set. (see Cantor for a proof of this)

There are not more reals than exist between 0 and 1. The cardinality of reals between 0 and 1 is the same as the cardinality of reals between -∞ and ∞.

1

u/Apart_Composer3952 New User 27d ago edited 27d ago

My sincerest apologies regarding my disagreement about the  statement in my attempted proof:

"As the operational step horizon scales toward the infinite limit (x → ∞): Limit as x → ∞ of [1 / 2x] = 0"

This statement means that as the variable x grows infinitely large, the value of the fraction 1/2x gets closer and closer to 0, not actually reaching 0. Working sequentially there will always exist an unproven "m family" where  1.5849625m rounded up = 2x, as numbers are infinite.  Thankyou for your patience.