On 3/13/2025 6:45 AM, WM wrote:
On 13.03.2025 01:43, Jim Burns wrote:
A single (lossless) exchange cannot delete an O
Finitely.many (lossless) exchanges cannot delete an O
>
Infinitely.many (lossless) exchanges can delete an O
>
No.
Your secret is that
your infiniteᵂᴹ and our infiniteⁿᵒᵗᐧᵂᴹ
mean different things.
Your "No" responds to infiniteᵂᴹ,
but I wrote infiniteⁿᵒᵗᐧᵂᴹ.
Infinite is not _only_ bigger. It's different.
>
It obeys logic or it is of no value.
Consider the value of infiniteⁿᵒᵗᐧᵂᴹ
Some preliminaries to clarify finiteⁿᵒᵗᐧᵂᴹ
It's good to have such things laid out clearly.
I don't expect you (WM) to object to these points,
but who knows.
⎛⎛ In my opinion, |A| would be
⎜⎜ harder to read than #A in
⎜⎜ this next little bit.
⎜⎝ I write #A = |A| = size of set A
⎜
⎜ For at least some sets,
⎜ fuller.by.one sets are larger.
⎜ #{1} < #{1,Bob}
⎜ #{1,2} < #{1,2,Bob}
⎜ #{unicorns} < #{unicorns.and.Bob}
⎜
⎜⎛ Cᣕᶜ is fuller.by.one than C
⎜⎜ Cᣕᶜ = C∪{c} ≠ C
⎜⎜ {1,Bob},{1,2,Bob},{unicorns.and.Bob}
⎜⎜ are examples of {1}ᣕᶜ,{1,2}ᣕᶜ,{unicorns}ᣕᶜ
⎜⎜
⎜⎜ Sets for which
⎜⎜ fuller.by.one sets are larger
⎜⎝ are finiteⁿᵒᵗᐧᵂᴹ.
⎜
⎜ Consider the set of set.sizes such that
⎜ fuller.by.one sets are larger.
⎜ Consider the set {#C:#C<#Cᣕᶜ}
⎜ {#C:#C<#Cᣕᶜ} is the set of finiteⁿᵒᵗᐧᵂᴹ set.sizes.
⎜
⎜ If #A < #Aᣕᵃ
⎜ (A has fuller.by.one Aᣕᵃ which are larger)
⎜ then set.size #A is in {#C:#C<#Cᣕᶜ}
⎜ and the other way 'round, too.
⎜ #A < #Aᣕᵃ ⇔ #A ∈ {#C:#C<#Cᣕᶜ}
⎜
⎜ For #A < #Aᣕᵃ
⎜ the size of the set of smaller set sizes = #A
⎜ #{#C:#C<#A<#Aᣕᵃ} = #A
⎜⎛ For example,
⎜⎝ #{Bob,Kevin} = 2 = #{0,1}
⎜
⎜ No subset is larger than its superset.
⎜ A ⊆ B ⇒ #A ≤ #B
⎜
⎜ For each #A ∈ {#C:#C<#Cᣕᶜ}
⎝ #A = #{#C:#C<#A<#Aᣕᵃ} ≤ #{#C:#C<#Cᣕᶜ})
Consider the value of infiniteⁿᵒᵗᐧᵂᴹ
⎛ A is smaller than B iff
⎜ fuller.by.one Aᣕᵃ is smaller than fuller.by.one Bᣕᵇ
⎜ #A < #B ⇔ #Aᣕᵃ < #Bᣕᵇ
⎜
⎜ Let B = Aᣕᵃ
⎜ #A < #Aᣕᵃ ⇔ #Aᣕᵃ < #Aᣕᵃᵇ
⎜ A is finiteⁿᵒᵗᐧᵂᴹ iff larger Aᣕᵃ is finiteⁿᵒᵗᐧᵂᴹ
⎝ There is no largest finiteⁿᵒᵗᐧᵂᴹ set.
⎛ #A ∈ {#C:#C<#Cᣕᶜ} ⇒
⎜ #A < #Aᣕᵃ ⇒
⎜ #Aᣕᵃ < #Aᣕᵃᵇ ⇒
⎜ #Aᣕᵃ ∈ {#C:#C<#Cᣕᶜ}
⎜
⎜ The set of finiteⁿᵒᵗᐧᵂᴹ sizes is inductive.
⎜ #A ∈ {#C:#C<#Cᣕᶜ} ⇒ #Aᣕᵃ ∈ {#C:#C<#Cᣕᶜ}
⎝ #{} ∈ {#C:#C<#Cᣕᶜ}
⎛ For each #A ∈ {#C:#C<#Cᣕᶜ}
⎜⎛ #A ≤ #{#C:#C<#Cᣕᶜ})
⎜⎜ #Aᣕᵃ ≤ #{#C:#C<#Cᣕᶜ})
⎜⎜ #A < #Aᣕᵃ ≤ #{#C:#C<#Cᣕᶜ}
⎜⎝ #A ≠ #{#C:#C<#Cᣕᶜ}
⎜
⎜ #{#C:#C<#Cᣕᶜ} ∉ {#C:#C<#Cᣕᶜ}
⎜ The set of finiteⁿᵒᵗᐧᵂᴹ set.sizes does not have
⎜ a finiteⁿᵒᵗᐧᵂᴹ set size.
⎜
⎜⎛ It seems as though that contradicts
⎜⎜ your (WM's) idea of a proof.by.induction.
⎜⎜
⎜⎜ It does not contradict
⎜⎜ a proof that a subset is the whole.set
⎜⎜ by proving that that subset is inductive,
⎜⎜ for a whole.set which is known to be
⎜⎜ its.own.only.inductive.subset,
⎜⎜ which is what's more commonly called
⎝⎝ a proof.by.induction.
⎛⎛ #{#C:#C<#Cᣕᶜ} ∉ {#C:#C<#Cᣕᶜ}
⎜⎜
⎜⎝ ¬(#{#C:#C<#Cᣕᶜ} < #{#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ)
⎜
⎜⎛ {#C:#C<#Cᣕᶜ} ⊆ {#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ
⎜⎜
⎜⎝ ¬(#{#C:#C<#Cᣕᶜ} > #{#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ)
⎜
⎝ #{#C:#C<#Cᣕᶜ} = #{#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ
Consider the value of infiniteⁿᵒᵗᐧᵂᴹ
{#C:#C<#Cᣕᶜ} is an infiniteⁿᵒᵗᐧᵂᴹ set.
As an infiniteⁿᵒᵗᐧᵂᴹ set,
{#C:#C<#Cᣕᶜ} is the same size as
fuller.by.one and emptier.by.one sets,
which is why
enough size.conserving swaps
can erase Bob from {#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ:
size is conserved
going from {#C:#C<#Cᣕᶜ}ᣕᴮᵒᵇ to {#C:#C<#Cᣕᶜ}
Date | Sujet | # | | Auteur |
12 Mar 25 | The existence of dark numbers proven by the thinned out harmonic series | 451 | | WM |
12 Mar 25 |  Re: The existence of dark numbers proven by the thinned out harmonic series | 450 | | Alan Mackenzie |
12 Mar 25 |   Re: The existence of dark numbers proven by the thinned out harmonic series | 449 | | WM |
12 Mar 25 |    The non-existence of "dark numbers" [was: The existence of dark numbers proven by the thinned out harmonic series] | 448 | | Alan Mackenzie |
12 Mar 25 |     Re: The non-existence of "dark numbers" [was: The existence of dark numbers proven by the thinned out harmonic series] | 444 | | WM |
12 Mar 25 |      Re: The non-existence of "dark numbers" | 414 | | Alan Mackenzie |
12 Mar 25 |       Re: The non-existence of "dark numbers" | 413 | | WM |
12 Mar 25 |        Re: The non-existence of "dark numbers" | 412 | | Alan Mackenzie |
12 Mar 25 |         Re: The non-existence of "dark numbers" | 6 | | Moebius |
13 Mar 25 |          Re: The non-existence of "dark numbers" | 1 | | WM |
13 Mar 25 |          Re: The non-existence of "dark numbers" | 4 | | Alan Mackenzie |
13 Mar 25 |           Re: The non-existence of "dark numbers" | 3 | | Moebius |
13 Mar 25 |            Re: The non-existence of "dark numbers" | 2 | | WM |
13 Mar 25 |             Re: The non-existence of "dark numbers" | 1 | | joes |
13 Mar 25 |         Re: The non-existence of "dark numbers" | 401 | | WM |
13 Mar 25 |          Re: The non-existence of "dark numbers" | 399 | | Alan Mackenzie |
13 Mar 25 |           Re: The non-existence of "dark numbers" | 397 | | WM |
13 Mar 25 |            Re: The non-existence of "dark numbers" | 3 | | joes |
13 Mar 25 |             Re: The non-existence of "dark numbers" | 2 | | WM |
14 Mar 25 |              Re: The non-existence of "dark numbers" | 1 | | joes |
13 Mar 25 |            Re: The non-existence of "dark numbers" | 393 | | Alan Mackenzie |
14 Mar 25 |             Re: The non-existence of "dark numbers" | 392 | | WM |
14 Mar 25 |              Re: The non-existence of "dark numbers" | 7 | | FromTheRafters |
14 Mar 25 |               Re: The non-existence of "dark numbers" | 6 | | WM |
14 Mar 25 |                Re: The non-existence of "dark numbers" | 5 | | FromTheRafters |
14 Mar 25 |                 Re: The non-existence of "dark numbers" | 4 | | WM |
15 Mar 25 |                  Re: The non-existence of "dark numbers" | 3 | | FromTheRafters |
15 Mar 25 |                   Re: The non-existence of "dark numbers" (thread too long, nothing in it) | 1 | | Ross Finlayson |
15 Mar 25 |                   Re: The non-existence of "dark numbers" | 1 | | WM |
14 Mar 25 |              Re: The non-existence of "dark numbers" | 383 | | Alan Mackenzie |
14 Mar 25 |               Re: The non-existence of "dark numbers" | 382 | | WM |
14 Mar 25 |                Re: The non-existence of "dark numbers" | 380 | | Alan Mackenzie |
14 Mar 25 |                 Re: The non-existence of "dark numbers" | 379 | | WM |
15 Mar 25 |                  Re: The non-existence of "dark numbers" | 371 | | Alan Mackenzie |
15 Mar 25 |                   Re: The non-existence of "dark numbers" | 370 | | WM |
15 Mar 25 |                    Re: The non-existence of "dark numbers" | 4 | | joes |
15 Mar 25 |                     Re: The non-existence of "dark numbers" | 3 | | WM |
15 Mar 25 |                      Re: The non-existence of "dark numbers" | 2 | | joes |
15 Mar 25 |                       Re: The non-existence of "dark numbers" | 1 | | WM |
15 Mar 25 |                    Re: The non-existence of "dark numbers" | 362 | | Alan Mackenzie |
15 Mar 25 |                     Re: The non-existence of "dark numbers" | 361 | | WM |
16 Mar 25 |                      Re: The non-existence of "dark numbers" | 356 | | Alan Mackenzie |
16 Mar 25 |                       Re: The non-existence of "dark numbers" | 355 | | WM |
16 Mar 25 |                        Re: The non-existence of "dark numbers" | 268 | | Jim Burns |
16 Mar 25 |                         Re: The non-existence of "dark numbers" | 267 | | WM |
16 Mar 25 |                          Re: The non-existence of "dark numbers" | 266 | | Jim Burns |
16 Mar 25 |                           Re: The non-existence of "dark numbers" | 265 | | WM |
16 Mar 25 |                            Re: The non-existence of "dark numbers" | 264 | | Jim Burns |
16 Mar 25 |                             Re: The non-existence of "dark numbers" | 263 | | WM |
17 Mar 25 |                              Re: The non-existence of "dark numbers" | 262 | | Jim Burns |
17 Mar 25 |                               Re: The non-existence of "dark numbers" | 261 | | WM |
17 Mar 25 |                                Re: The non-existence of "dark numbers" | 260 | | Jim Burns |
17 Mar 25 |                                 Re: The non-existence of "dark numbers" | 259 | | WM |
17 Mar 25 |                                  Re: The non-existence of "dark numbers" | 258 | | Jim Burns |
18 Mar 25 |                                   Re: The non-existence of "dark numbers" | 257 | | WM |
18 Mar 25 |                                    Re: The non-existence of "dark numbers" | 256 | | Jim Burns |
18 Mar 25 |                                     Re: The non-existence of "dark numbers" | 255 | | WM |
19 Mar 25 |                                      Re: The non-existence of "dark numbers" | 254 | | Jim Burns |
19 Mar 25 |                                       Re: The non-existence of "dark numbers" | 253 | | WM |
19 Mar 25 |                                        Re: The non-existence of "dark numbers" | 252 | | Jim Burns |
20 Mar 25 |                                         Re: The non-existence of "dark numbers" | 251 | | WM |
20 Mar 25 |                                          Re: The non-existence of "dark numbers" | 250 | | Jim Burns |
20 Mar 25 |                                           Re: The non-existence of "dark numbers" | 249 | | WM |
20 Mar 25 |                                            Re: The non-existence of "dark numbers" | 248 | | Jim Burns |
21 Mar 25 |                                             Re: The non-existence of "dark numbers" | 247 | | WM |
21 Mar 25 |                                              Re: The non-existence of "dark numbers" | 246 | | Jim Burns |
21 Mar 25 |                                               Re: The non-existence of "dark numbers" | 245 | | WM |
21 Mar 25 |                                                The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"] | 183 | | Alan Mackenzie |
21 Mar 25 |                                                 Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"] | 40 | | Moebius |
21 Mar 25 |                                                  Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"] | 37 | | Moebius |
21 Mar 25 |                                                   Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"] | 2 | | Moebius |
21 Mar 25 |                                                    Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"] | 1 | | Moebius |
21 Mar 25 |                                                   Re: The reality of sets, on a scale of 1 to 10 | 34 | | Alan Mackenzie |
21 Mar 25 |                                                    Re: The reality of sets, on a scale of 1 to 10 | 32 | | Moebius |
22 Mar 25 |                                                     Re: The reality of sets, on a scale of 1 to 10 | 1 | | Ross Finlayson |
22 Mar 25 |                                                     Re: The reality of sets, on a scale of 1 to 10 | 29 | | Ralf Bader |
22 Mar 25 |                                                      Re: The reality of sets, on a scale of 1 to 10 | 28 | | Moebius |
22 Mar 25 |                                                       Re: The reality of sets, on a scale of 1 to 10 | 2 | | Moebius |
22 Mar 25 |                                                        Re: The reality of sets, on a scale of 1 to 10 | 1 | | Moebius |
23 Mar 25 |                                                       Re: The reality of sets, on a scale of 1 to 10 | 25 | | Ross Finlayson |
23 Mar 25 |                                                        Re: The reality of sets, on a scale of 1 to 10 | 24 | | Jim Burns |
23 Mar 25 |                                                         Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 23 | | Ross Finlayson |
24 Mar 25 |                                                          Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 19 | | Chris M. Thomasson |
24 Mar 25 |                                                           Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 18 | | Jim Burns |
24 Mar 25 |                                                            Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 11 | | Ross Finlayson |
24 Mar 25 |                                                             Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 10 | | Jim Burns |
25 Mar 25 |                                                              Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 9 | | Ross Finlayson |
25 Mar 25 |                                                               Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 3 | | Jim Burns |
25 Mar 25 |                                                                Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 2 | | Ross Finlayson |
25 Mar 25 |                                                                 Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 1 | | Jim Burns |
25 Mar 25 |                                                               Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 5 | | Jim Burns |
25 Mar 25 |                                                                Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 4 | | Ross Finlayson |
25 Mar 25 |                                                                 Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 3 | | Jim Burns |
25 Mar 25 |                                                                  Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 2 | | Ross Finlayson |
25 Mar 25 |                                                                   Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 1 | | Jim Burns |
26 Mar 25 |                                                            Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 6 | | Chris M. Thomasson |
27 Mar 25 |                                                             Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 5 | | Jim Burns |
27 Mar 25 |                                                              Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 4 | | FromTheRafters |
27 Mar 25 |                                                               Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 1 | | Jim Burns |
27 Mar 25 |                                                               Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 2 | | Ross Finlayson |
27 Mar 25 |                                                                Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 1 | | Ross Finlayson |
24 Mar 25 |                                                          Re: The reality of sets, on a scale of 1 to 10 (theory of theories) | 3 | | Jim Burns |
22 Mar 25 |                                                     Re: The reality of sets, on a scale of 1 to 10 | 1 | | WM |
22 Mar 25 |                                                    Re: The reality of sets, on a scale of 1 to 10 | 1 | | WM |
22 Mar 25 |                                                  Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"] | 2 | | WM |
22 Mar 25 |                                                 Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"] | 142 | | WM |
21 Mar 25 |                                                Re: The non-existence of "dark numbers" | 3 | | FromTheRafters |
22 Mar 25 |                                                Re: The non-existence of "dark numbers" | 58 | | Jim Burns |
16 Mar 25 |                        Re: The non-existence of "dark numbers" | 85 | | Alan Mackenzie |
16 Mar 25 |                        Re: The non-existence of "dark numbers" | 1 | | joes |
16 Mar 25 |                      Re: The non-existence of "dark numbers" | 4 | | joes |
15 Mar 25 |                    Re: The non-existence of "dark numbers" | 3 | | Chris M. Thomasson |
15 Mar 25 |                  Re: The non-existence of "dark numbers" | 7 | | joes |
14 Mar 25 |                Re: The non-existence of "dark numbers" | 1 | | joes |
14 Mar 25 |              Re: The non-existence of "dark numbers" | 1 | | joes |
14 Mar 25 |           Re: The non-existence of "dark numbers" | 1 | | Chris M. Thomasson |
13 Mar 25 |          Re: The non-existence of "dark numbers" | 1 | | joes |
13 Mar 25 |         Re: The non-existence of "dark numbers" | 4 | | Ben Bacarisse |
12 Mar 25 |      Re: The non-existence of "dark numbers" [was: The existence of dark numbers proven by the thinned out harmonic series] | 29 | | Jim Burns |
12 Mar 25 |     Re: The non-existence of "dark numbers" [was: The existence of dark numbers proven by the thinned out harmonic series] | 2 | | FromTheRafters |
12 Mar 25 |     Re: The non-existence of "dark numbers" [was: The existence of dark numbers proven by the thinned out harmonic series] | 1 | | Jim Burns |