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Topic: 1.17 - Is it sensible to link cardinality of two sets with a bijection?
Replies: 7   Last Post: Jun 7, 2014 3:46 AM

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Posts: 1,361
Registered: 11/1/13
1.17 - Is it sensible to link cardinality of two sets with a bijection?
Posted: Jun 6, 2014 4:42 AM
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A small investigation of the following applet, will reveal that this is an absurd idea:

The intervals (0,1) and (0.5, 0.75) are mapped using the bijective function RtoI() and its inverse ItoR().

Never mind that the finite line segments are of lengths 1 and 0.25 respectively, but idiot Cantorites imagine in their dysfunctional minds, that these two intervals have the same number of elements, that is, numbers.

However, the problem is that not every output of the bijection is a rational number! In most cases, it is an incommensurable magnitude.

So one has:

numbers to numbers
numbers to incommensurable magnitudes
incommensurable magnitudes to numbers
incommensurable magnitudes to incommensurable magnitudes

Each set consists of two types of elements: numbers and incommensurable magnitudes. This is in total violation of the definition of set, which states that elements must be distinct and of the same type. While numbers exist as distinct elements, incommensurable magnitudes DO NOT exist as distinct elements.

For example, what is pi or sqrt(2)? The whole idea of bijection implying same cardinality is therefore absurd!

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