20220928, 07:13  #34 
Romulan Interpreter
"name field"
Jun 2011
Thailand
2·47·109 Posts 

20221002, 10:20  #35 
"Anonymous"
Sep 2022
finding m52
9_{16} Posts 
(1+1+1+1+1+1+...) is indeed less than (1+2+3+4+5+6+...)
In the Gauss formula, there is a pattern regarding powers of 10.
1  10 = 11 x 5 = 55 1  100 = 101 x 50 = 5050 1  1000 = 1001 x 500 = 500500 The pattern: 1  x = (x+1)*(x/2) = 5 followed by ((log10 of x)1 zeroes) followed by 5 followed by ((log 10 of x)1 zeroes) See how there are ((log10 of x)1)*2 zeroes in the final result? Therefore: 1  ∞ = (∞+1)*(∞/2) = 5 followed by (∞1 zeroes) followed by 5 followed by (∞1 zeroes). Obviously, (1+1+1+1+1+1+...) results in ∞, noticeably less than the result of (1+2+3+4+5+6+...). 
20221002, 11:01  #36  
Sep 2002
Database er0rr
29×151 Posts 
Quote:
Last fiddled with by paulunderwood on 20221002 at 11:06 

20221002, 11:56  #37 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
19F2_{16} Posts 

20221002, 14:15  #38 
Feb 2017
Nowhere
17D8_{16} Posts 
If you actually make it welldefined, of course, you spoil all the fun. First, you have to be clear about what kind of number you mean by . There are cardinal ("how many") numbers and ordinal numbers (the ordinal types of sets with wellorderings)  first, second, third, etc.
I'll assume we're dealing with cardinal numbers, and is the cardinal number of the positive integers. Then all the indicated infinities are . Finite ordinal numbers are of the form {1, 2, ... n} (positive integers from 1 to n) with the usual linear ordering. Infinite ordinal numbers are much more complicated. The first infinite ordinal number is the ordinal type of the positive integers with the usual ordering, which may also be viewed as the ordinal type of the set of all finite ordinals, ordered by one being a section (initial segment) of another. The cardinality of this set is not finite, because no ordinal number can be a section of itself. This cardinality is . The ordinal type is usually denoted . Defining ordinal addition by concatenation, we see that , but . The ordinal type of the set of all ordinal types which are either finite or of sets with cardinality is usually denoted . The cardinality of the set of such ordinals is the next infinite cardinal number after , denoted . Last fiddled with by Dr Sardonicus on 20221002 at 14:41 Reason: Insert omitted word 
20221004, 13:57  #39  
"Anonymous"
Sep 2022
finding m52
3^{2} Posts 
Quote:
Note: See added emphasis in quote. Last fiddled with by Dr Sardonicus on 20221004 at 14:20 Reason: As indicated 

20221009, 02:00  #40 
"Anonymous"
Sep 2022
finding m52
11_{8} Posts 
