# Goodstein’s theorem

[guess]

Goodstein’s theorem is an example that sometimes a finite result requires the existence of infinity in its proof.

— Me@2021-05-09 11:06:34 PM

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Goodstein’s theorem itself assumes that there is an infinite number of natural numbers, so it is not really a finite result.

— Me@2017-02-20 06:16:28 PM

[guess]

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# Universal wave function, 21

For all, 9

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A problem of universal wave function (universe) is that universe is a relative concept.

Another problem is that wave function is also.

— Me@2017-05-10 05:46:44 PM

— Me@2021-04-09 06:25:07 PM

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universe ~ 100%

But 100% of what?

— Me@2021-04-09 05:20:23 PM

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The wave function is expressed in terms of basis state vectors.

So it will have a different form if you choose a different basis.

— Me@2021-04-09 06:29:20 PM

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# Omnipotence 4.2

When responding to the question “can X create a stone that it cannot lift”, another flawed argument is

X can create the stone that it cannot lift but it chooses not to create it. So there is no stone it cannot lift yet. So X has not failed the omnipotence test.

This argument is wrong.

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When we ask “can X choose to create a stone that it cannot lift”, we are discussing whether X has an ability. When we discuss ability, it is always about a potential, a possibility.

Y is able to do action B

always means that

“Y does B” is possible,

which is equivalent to

“Y does B” is not contradictory to any logical laws nor physical laws.

“Whether Y has already done B or will do B” is not the point.

.

If we allow such “Y can do B but it chooses not to” argument, then anyone is omnipotent. For example,

Can you fly?

I can fly but I choose not to. So even though you have never seen me flying and will never see me flying, it is not because I cannot fly; it is just because I choose not to.

Can you choose to fly?

I can choose to fly but I choose not to choose to fly.

This type of arguments make the word “can“ meaningless.

— Me@2020-03-30 06:52:58 AM

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# Omnipotence 4.1

For all, 3 | Omnipotence

For all, 3.2 | Omnipotence 2

You can find them by searching “omnipotence” using this blog’s search box.

— Me@2020-04-08 03:17:34 PM

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If X is omnipotent, X can create a stone that it cannot lift. Then X is not omnipotent, because there is a stone it cannot lift. So omnipotence is a self-contradictory concept.

What if we define omnipotence not as “being able to do anything” but as “being able to do anything except logical self-contradictory ones“?

In order words, omnipotence means that being able to do anything logically possible. Omnipotence does not mean that being able to do also logically impossible things.

This re-definition is not useful, because the original meaning of “being omnipotent” already is “being able to do anything except logical self-contradictory ones“.

There is no re-definition needed. You can only say that the re-definition clarifies the original meaning of “being omnipotent”. However, this clarification cannot eliminate the self-contradictory nature of the meaning of “omnipotence” itself. For example, the following argument is wrong.

If X is omnipotent, “X can create a stone that it cannot lift” is self-contradictory because it is contradictory to “X is omnipotent”.

Since “X can create a stone that it cannot lift” is logically impossible, it should not be a requirement of being omnipotent.

This argument is wrong because:

1. “X can create a stone that it cannot lift” is not SELF-contradictory.

2. “X can create a stone that it cannot lift” is not logically impossible, because, for example, even a human being can create an object that he cannot lift. For example, human beings can create a car that no single person can lift.

Then someone might keep arguing that

But if X is omnipotent, “X can create a stone that it cannot lift” means that “X is omnipotent and X can create a stone it cannot lift”, which is logically impossible. So “X cannot create a stone that it cannot lift” does not make X non-omnipotent.

In other words, “whether X can create a stone that it cannot lift” should not be the requirement of the omnipotence test.

The argument is wrong, because what we are questioning is

Can someone X be omnipotent?

or

Is omnipotence logically possible?

.

Remember:

“Being logically possible” means “not self-contradictory”.

.

If “X is omnipotent” is true,

then “X can create a stone that it cannot lift” is true.

Then “there is a stone that X cannot lift” is true.

Then “X is not omnipotent” is true.

But “X is not omnipotent” is contradictory to the assumption “X is omnipotent“.

So “X is omnipotent” is self-contradictory.

So the question “whether an entity X can be omnipotent and create a stone that it cannot lift” is illegitimate because “an entity X is omnipotent” is logically impossible in the first place. It should not be placed within a question.

Note that our omnipotent test is

“whether an entity X can create a stone that it cannot lift”,

NOT “whether an entity X can be omnipotent and create a stone that it cannot lift”,

NOR “whether an omnipotent entity X can create a stone that it cannot lift”.

— Me@2020-03-30 06:52:58 AM

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# 機遇再生論 1.8

$P(A_{10,000,000})$
$\approx 1.2398 \times 10^{-61}$

$P(A_{20,000,000})$
$\approx 1.2398 \times 10^{-54}$

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（問：那我不需要在「洗完一千萬次牌後，發現原本排列 A 還未重新出現時」，才問_另一個_問題，因為，事先透過運算，就已經知道，那機會十分之微。

」）

$P(A_{10,000,000^{10}}) = 0.9999999...$

.

（問：為什麼要「相對於現在的你而言」？）

.

$P(A_{10,000,000^{10}})$ 了，

$P(A_{10,000,000^{10} - 1})$

（問：你的意思是，即使我洗了（例如）一千萬牌，仍然得不回原本的排列 A，只要我洗多一千萬次，得回 A 的機會，就會大一點？）

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「機遇再生論」在同情地理解下，可以有這個意思。

— Me@2018-03-20 02:26:35 PM

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# 機遇再生論 1.7

$P(A_m)= 1 - (1 - \frac{1}{N})^m$

$P(A_m)$
$= 1 - (1 - \frac{1}{N})^m$
$= 1 - (1 - \frac{1}{52!})^{10,000,000}$

$1.239799930857148592 \times 10^{-61}$

.

「（在一次牌都未洗的時候問）洗牌 二千萬 次，起碼一次洗到原本排列 A 的機會率」

「（在一次牌都未洗的時候問）洗牌 一千萬 次，起碼一次洗到原本排列 A 的機會率」。

.

$P(A_m)$
$= 1 - (1 - \frac{1}{N})^m$
$= 1 - (1 - \frac{1}{52!})^{10,000,000 \times 2}$
$\approx 2.479599861714297185 \times 10^{-61}$,

$P(A_m)$
$= 1 - (1 - \frac{1}{52!})^{10,000,000 \times 10,000,000}$
$\approx 1.2397999308571485923950342 \times 10^{-54}$

$m = 10,000,000^3, P(A_m) = 1 - (1 - \frac{1}{52!})^{10,000,000^3} \approx 1.2398 \times 10^{-47}$

$m = 10,000,000^4, P(A_m) \approx 1.2398 \times 10^{-40}$

$m = 10,000,000^8, P(A_m) \approx 1.2398 \times 10^{-12}$

$m = 10,000,000^9, P(A_m) \approx 0.000012398$

$m = 10,000,000^{10}, P(A_m) = 0.9999999...$

.

$P(A_{10,000,000^{10}}) = 0.9999999...$

.

（問：你的意思是，即使我洗了（例如）一千萬牌，仍然得不回原本的排列 A，只要我洗多一千萬次，得回 A 的機會，就會大一點？）

.

$P(A) = \frac{1}{N}$

$(N = 52! \approx 8.07 \times 10^{67})$

$P(A_{10,000,000})$
$\approx 1.2398 \times 10^{-61}$

$P(A_{10,000,000})$
$\approx 1.2398 \times 10^{-61}$

$P(A_{20,000,000})$
$\approx 1.2398 \times 10^{-54}$

— Me@2018-02-23 08:21:52 PM

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# 機遇再生論 1.6

.

（而這個意思，亦在「機遇再生論」的原文中，用作其理據。）

$P(A) = \frac{1}{N}$

$P(\text{not} A) = 1 - \frac{1}{N}$

.

$P(A) = \frac{1}{N}$

$P(\text{not} A) = 1 - \frac{1}{N}$

.

(問：那樣，為什麼要問多一次呢？）

「如果洗牌兩次，起碼一次洗到原本排列 A 的機會率是多少？」

$A_2$ = 兩次洗牌的結果，起碼一次洗到原本排列 A

$A_2$ 的互補事件為「不是 $A_2$」：

= 兩次洗牌的結果，不是起碼一次洗到原本排列 A

= 兩次洗牌的結果，都不是排列 A

$P(\text{not} A_2) = (1 - \frac{1}{N})^2$

$P(A_2)$
$= 1 - P(\text{not} A_2)$
$= 1 - (1 - \frac{1}{N})^2$

.

$P(A_m)= 1 - (1 - \frac{1}{N})^m$

$P(A_m)$
$= 1 - (1 - \frac{1}{N})^m$
$= 1 - (1 - \frac{1}{52!})^{10,000,000}$

$1.239799930857148592 \times 10^{-61}$

— Me@2018-01-25 12:38:39 PM

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# 機遇再生論 1.5

（請參閱本網誌，有關「重言句」、「經驗句」和「印證原則」的文章。）

「同情地理解」的意思是，有些理論，雖然在第一層次的分析之後，有明顯的漏洞，但是，我們可以試試，代入作者發表該理論時的，心理狀態和時空情境；研究作者發表該理論的，緣起和動機；從而看看，該理論不行的原因，會不會只是因為，作者的語文或思考不夠清晰，表達不佳而已？

（而這個意思，亦在「機遇再生論」的原文中，用作其理據。）

$P(A) = \frac{1}{N}$

$P($not $A) = 1 - \frac{1}{N}$

— Me@2017-12-18 02:51:11 PM

# Determined by what?

If you say “an event is determined”, in order to be meaningful, you have to specify, explicitly or by context, that the event is determined by whom.

Similarly, if you say something is free, you have to specify “free from what” or “free with respect to what”.

free ~ independent of

Without a grammatical object, the phrase “independent of” is meaningless, unless the context has implied what that grammatical object is.

— Me@2015-05-23

free [without an object] ~ free from everything

is meaningless, because the word “everything” is meaningful only if it has a context.

— Me@2017-07-20

# The meanings of ONE

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One bag of apples, one apple, one slice of apple — which of these is one unit? Explore the basic unit of math (explained by a trip to the grocery store!) and discover the many meanings of one.

— Lesson by Christopher Danielson, animation by TED-Ed

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A unit ~ a definition of one

(cf. One is one … or is it? — TED-Ed)

— Me@2017-02-13 8:48 AM

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One is not a number, in the following sense:

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Primality of one

Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be a prime number. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer’s list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to arrive at the consensus that 1 is not a prime number, but rather forms its own special category as a “unit”.

A large body of mathematical work would still be valid when calling 1 a prime, but Euclid’s fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler’s totient function or the sum of divisors function.

— Wikipedia on Prime number

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As long as something exists, it is possible to define one.

One as the basis for counting (number); one itself is not a number, in the sense that one is for existence, not for counting.

When counting, we have to know count with respect to what. That “what” is a “unit”, aka one.

That is why

x times 1 = x

— Me@2017-02-13 8:48 AM

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$\displaystyle{2 \times 1 = 2}$

there are 2 units of apple == there are 2 apples

— Me@2021-08-22 07:13:46 AM

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$\displaystyle{2 \times 1 = 2}$

2 units = 2

— Me@2021-08-22 10:27:52 AM

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You define one in a context.

But you cannot define two without defining one.

— Me@2017-02-14 07:10:51 AM

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You define a unit in a context.

But you cannot define a number without defining a unit.

— Me@2021-08-21 10:08:58 PM

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# 注定外傳 2.6

Can it be Otherwise? 2.6 | The Beginning of Time, 7.3

『所有』，就是『場所之有』。

— Me@2016-05-18 11:40:31 AM

# 注定外傳 2.5

Can it be Otherwise? 2.5 | The Beginning of Time, 7.2

4. 即使可以追溯到「時間的起點」（第一因），所謂的「可以」，只是宏觀而言，決不會細節到可以推斷到，你有沒有自由，明天七時起牀。

（問：如果因果環環緊扣，即使細節不完全知道，至少理論上，我們可以知道，如果「第一因」本身有自由，那其他個別事件，就有可能有（來自「第一因」的）自由；如果連「第一因」也沒有自由，那其他個別事件，都一律沒有自由。

「第一因有自由。」

「第一因」根據定義，是沒有原因的。亦即是話，「時間的起點」，再沒有「之前」。而「有自由」，就即是「有其他可能性」。所以，「第一因有自由」的意思是，

「第一因還有其他的可能性。」

（問：如果有「造物主」，祂不就是那個誰，可以從宇宙之初的不同可能性中，選擇一個去實現嗎？）

「因果是否真的『環環緊扣』，有沒有可能，有『同因不同果』的情況？」

— Me@2016-03-15 08:43:58 AM

# 注定外傳 2.3

Can it be Otherwise? 2.3

— Me@2016-01-06 03:17:54 PM

# 注定外傳 2.2

Can it be Otherwise? 2.2

「成績注定」和「主動溫習」，根本沒有矛盾。

— Me@2015-12-29 03:12:39 PM

# 注定外傳 2.1.2

Can it be Otherwise? 2.1.2 | The problem of induction 2.2

1. 當你的「相似事件」和「原本事件」的結果相同時，你只可以知道「原本事件」，可能是注定；你並不可以肯定「原本事件」，一定是注定，因為，你並不能保證，下一件「相似事件」的結果，會不會仍然和「原本事件」相同。

2. 當你的「相似事件」和「原本事件」的結果不同時，你亦不可以肯定「原本事件」，一定是偶然，因為，結果不同，可能只是由於「相似事件」和「原本事件」，不夠相似而已。

— Me@2015-11-17 02:02:03 PM

# 注定外傳 2.1.1

Can it be Otherwise? 2.1.1 | The problem of induction 2.1

（層次一的事件描述：）

（層次一的反證：）

（層次二 —— 準確一點的事件描述：）

（層次二 —— 詳細一點的反證：）

（層次三：）

（層次四：）

— Me@2015-11-17 02:02:03 PM

# 注定外傳 1.11

Can it be Otherwise? 1.11

— Me@2015-10-29 03:10:19 PM

Q: Can it be otherwise?

A: What is “it”?

— Me@2015-10-29 03:10:14 PM

# 注定外傳 1.10

Can it be Otherwise? 1.10

（問：如果只是「類似」，當然可以有不同結果。你應該直接問：

』）

（問：不是呀。在量子力學中，即使有兩組百分百一樣的物理系統，即使它們獲得完全相同的輸入，都可能有不同的輸出。）

「相同」的意思，並不是指「沒有可能找到任何分別」。

「相同」的意思是「分別小到不易察覺」。

」，

— Me@2015-10-29 10:12:16 PM

# 注定外傳 1.9

Can it be Otherwise? 1.9

（這裡的「東西」，是指宏觀的物理系統。至於兩粒微觀粒子，則有可能「全同」。但那是另一個話題，容後再談。）

（問：那如果連位置都相同呢？）

— Me@2015-10-07 02:52:21 PM