Proof by Induction with 2 Variables

[Dan Christensen]

Recall: PROOF BY INDUCTION FOR 1 VARIABLE

Suppose you want to prove that, for all m in N, we have P(m) being true.

It will be sufficient to prove:

1. P(0)

2. For all k in N, we have: P(k) => P(k+1)


*****************************************************************************

PROOF BY INDUCTION FOR 2 VARIABLES

Suppose you want to prove that, for all m, n in N, we have P(m, n) being true.

It will be sufficient to prove:

1. P(0,0)

2. For all k in N, we have: P(0, k) => P(0, k+1)

3. For all j, k in N, we have: P(j, k) => P(j+1, k)


Your comments?


Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

[FredJeffries]

On Monday, October 19, 2020 at 8:05:26 AM UTC-7, Dan Christensen wrote:
> Recall: PROOF BY INDUCTION FOR 1 VARIABLE
>
> Suppose you want to prove that, for all m in N, we have P(m) being true.
>
> It will be sufficient to prove:
>
> 1. P(0)
>
> 2. For all k in N, we have: P(k) => P(k+1)
>
>
> *****************************************************************************
>
> PROOF BY INDUCTION FOR 2 VARIABLES
>
> Suppose you want to prove that, for all m, n in N, we have P(m, n) being true.
>
> It will be sufficient to prove:
>
> 1. P(0,0)
>
> 2. For all k in N, we have: P(0, k) => P(0, k+1)
>
> 3. For all j, k in N, we have: P(j, k) => P(j+1, k)
>
>
> Your comments?

Congratulations. You have just proved that for all natural numbers j, k, the product j*k = 0

[Dan Christensen]

Think again.

[Me]

On Monday, October 19, 2020 at 5:05:26 PM UTC+2, Dan Christensen wrote:
>
> Recall: PROOF BY INDUCTION FOR 1 VARIABLE
>
> Suppose you want to prove that, for all m in N, we have P(m) being true.
>
> It will be sufficient to prove:
>
> 1. P(0)
>
> 2. For all k in N, we have: P(k) => P(k+1)
>
>
> *****************************************************************************
>

> Suppose you want to prove that, for all n, m in N, we have P(n, m) being true.

Let n0 an arbitrary element in IN.

> [Now] it will be sufficient to prove:

> 1. P(n0,0)
>
> 2. For all k in N, we have: P(n0, k) => P(n0, k+1)

Hence we get (by induction):

For all m in N: P(n0, m) .

Since n0 was an arbitrary element in N we get:

For all n in IN: For all m in N: P(n, m) .

[Me]

On Monday, October 19, 2020 at 5:25:32 PM UTC+2, Dan Christensen wrote:

> Think again.

We have to show:

> 1. P(0,0)
>
> 2. For all k in N, we have: P(0, k) => P(0, k+1)

Hence we may derive by induction

Ak e IN: P(0, k) .

Now assume that for some (arbitrary) j0 e IN:

Ak e IN: P(j0, k)

holds, and that we can prove

Ak e IN: P(j0+1, k)

Hence we get (by CP and generalization):

Aj e IN: (Ak e IN: P(j, k)) -> (Ak e IN: P(j+1, k)) (*)

Then (again by induction):

Aj e IN: Ak e IN: P(j, k) .

Now let's assume that after step 2. we can show:

> 3. For all j, k in N, we have: P(j, k) => P(j+1, k)

i.e. Aj e IN: Ak e IN: P(j, k) -> P(j+1, k)

Then a logical theorem would allow to derive

Aj e IN: (Ak e IN: P(j, k)) -> (Ak e IN: P(j+1, k)) .

[ since Ax(Fx -> Gx) |- AxFx -> AxGx ]

So we would get (*) again. Justifying an "application" of induction.

So it seems to me that your approach is correct.

Is there an error in my argument?

[Me]

On Monday, October 19, 2020 at 5:22:58 PM UTC+2, FredJeffries wrote:

> Congratulations. You have just proved that for all natural numbers j, k, the
> product j*k = 0

Are you sure? Could you please elaborate? I mean, write out your argument in detail?

[Mostowski Collapse]

Usually this here is proof by course of value
induction for two variables:

p(0,0)

forall x,y (forall z,t <z,t> < <x,y> => p(z,t)) => p(x,y))

Where:

<z,t> < <x,y> :<=> z < x v (z = x & t < y)

Its based on the well ordering of omega^2. Try
reducing it to two omega course of value inductions.

Can you? Can you reduce it to your schema?

[Mostowski Collapse]

You can drop p(0,0), and we need also write
the conclusion, so it would be:

forall x,y (forall z,t <z,t> < <x,y> => p(z,t)) => p(x,y))

=> forall u,v p(u,v)

[Dan Christensen]

On Monday, October 19, 2020 at 11:43:35 AM UTC-4, Me wrote:
> On Monday, October 19, 2020 at 5:05:26 PM UTC+2, Dan Christensen wrote:
> >
> > Recall: PROOF BY INDUCTION FOR 1 VARIABLE
> >
> > Suppose you want to prove that, for all m in N, we have P(m) being true.
> >
> > It will be sufficient to prove:
> >
> > 1. P(0)
> >
> > 2. For all k in N, we have: P(k) => P(k+1)
> >
> >
> > *****************************************************************************
> >
> > Suppose you want to prove that, for all n, m in N, we have P(n, m) being true.
>
> Let n0 an arbitrary element in IN.
>
> > [Now] it will be sufficient to prove:
>
> > 1. P(n0,0)

I think you will need P(0, 0) to prove this for arbitrary n0.

> >
> > 2. For all k in N, we have: P(n0, k) => P(n0, k+1)
>
> Hence we get (by induction):
>
> For all m in N: P(n0, m) .
>
> Since n0 was an arbitrary element in N we get:
>
> For all n in IN: For all m in N: P(n, m) .

[Dan Christensen]

On Monday, October 19, 2020 at 1:40:18 PM UTC-4, Dan Christensen wrote:
> On Monday, October 19, 2020 at 11:43:35 AM UTC-4, Me wrote:
> > On Monday, October 19, 2020 at 5:05:26 PM UTC+2, Dan Christensen wrote:
> > >
> > > Recall: PROOF BY INDUCTION FOR 1 VARIABLE
> > >
> > > Suppose you want to prove that, for all m in N, we have P(m) being true.
> > >
> > > It will be sufficient to prove:
> > >
> > > 1. P(0)
> > >
> > > 2. For all k in N, we have: P(k) => P(k+1)
> > >
> > >
> > > *****************************************************************************
> > >
> > > Suppose you want to prove that, for all n, m in N, we have P(n, m) being true.
> >
> > Let n0 an arbitrary element in IN.
> >
> > > [Now] it will be sufficient to prove:
> >
> > > 1. P(n0,0)
>
> I think you will need P(0, 0) to prove this for arbitrary n0.
>

I have formally proven (133 lines in DC Proof):

P(0,0)
& ALL(b):[b in n => [P(0,b) => P(0,s(b))]]
& ALL(b):[b in n => ALL(a):[a in n => [P(a,b) => P(s(a),b)]]]

=> ALL(a):ALL(b):[a in n & b in n => P(a,b)]

[FredJeffries]

No. I typed too soon. Sincere apologies.

[Dan Christensen]

On Monday, October 19, 2020 at 2:46:36 PM UTC-4, Archimedes Plutonium wrote:
> Logic Shitheads like AP...

WARNING TO STUDENTS: Don't be a victim of AP's fake math and science

AP is a malicious troll who really, REALLY wants you to fail in school just like he must have so long ago (in the 60's?). Then he would like to recruit you to his sinister Atom God Cult of Failure. Think I'm making this up? IN HIS OWN WORDS:


AP's fake math that can only be designed to promote failure in schools:

“Primes do not exist, because the set they were borne from has no division.”
--June 29, 2020 *** NEW ***

“The last and largest finite number is 10^604.”
--June 3, 2015

“0 appears to be the last and largest finite number”
--June 9, 2015

“0/0 must be equal to 1.”
-- June 9, 2015

“0 is an infinite irrational number.”
--June 28, 2015

“No negative numbers exist.”
--December 22, 2018

“Rationals are not numbers.”
--May 18, 2019

“The value of sin(45 degrees) = 1.” (Actually, sin(45 degrees) = 0.707. tan(45 degrees) = 1.)
--May 31, 2019

AP deliberately and repeatedly presented the truth table for OR as the truth table for AND:

“New Logic
AND
T & T = T
T & F = T
F & T = T
F & F = F”
--November 9, 2019

According to AP's “chess board math,” an equilateral triangle is right-triangle.
--December 11, 2019


AP seeks aid of Russian agents to promote failure in schools:

"Please--Asking for help from Russia-- russian robots-- to create a new, true mathematics [sic]"
--November 9, 2017


And if that wasn't weird enough...


AP's sinister Atom God Cult

“The totality, everything that there is [the universe], is only 1 atom of plutonium [Pu]. There is nothing outside or beyond this one atom of plutonium.”
--April 4, 1994

“The Universe itself is one gigantic big atom.”
--November 14, 2019

“Since God-Pu is marching on.
Glory! Glory! Atom Plutonium!
Its truth is marching on.
It has sounded forth the trumpet that shall never call retreat;
It is sifting out the hearts of people before its judgment seat;
Oh, be swift, my soul, to answer it; be jubilant, my feet!
Our God-Pu is marching on.”
--December 15, 2018 (Note: Pu is the atomic symbol for plutonium)

[Mostowski Collapse]

Where? And what did you use?

BTW: Interesting challenge. How compare different
induction principles. Lets say we have now:

- Dan-O-Matics: omega twice
- The usual omega^2

How do we compare the two? omega^2 induction is
for example used in cut-elimination.

Can we make a case of a predicate P, which is
not amenable to Dan-O-Matics schema, but to the
usual schema?

The problem is if forall u,v P(u,v) holds, then
also P(0,0) holds, and P(0,b)=>P(0,b') holds, and
P(a,b)=>P(a',b) holds.

Same for the second principle omega^2. if forall
u,v P(u,v) holds, then forall x,y (forall z,t <z,t> <
<x,y> => p(z,t)) => p(x,y)) holds as well.

So how are induction principles compared?

[Archimedes Plutonium]

Logic Shitheads like Dan Christensen, Jan Burse, Kibo Parry Moroney do not belong in the science newsgroups with their 10 OR 2 = 12, their 10 AND 2 being subtraction, their 938 is 12% short of 945. Throw the stalking shitheads out of sci.math.

5th published book

Suspend all College Classes in Logic, until they Fix their Errors // Logic series, book 1 Kindle Edition
by Archimedes Plutonium (Author)

First comes Logic-- think straight and clear which many logic and math professors are deaf dumb and blind to, and simply refuse to recognize and fix their errors.

The single biggest error of Old Logic of Boole and Jevons was their "AND" and "OR" connectors. They got them mixed up and turned around. For their logic ends up being that of 3 OR 2 = 5 with 3 AND 2 = 1, when even the local village idiot knows that 3 AND 2 = 5 (addition) with 3 OR 2 = either 3 or 2 (subtraction). And secondly, their error of the If->Then conditional. I need to make it clear enough to the reader why the true Truth Table of IF --> Then requires a U for unknown or uncertain with a probability outcome for F --> T = U and F --> F = U. Some smart readers would know that the reason for the U is because without the U, Logic has no means of division by 0 which is undefined in mathematics. You cannot have a Logic that is less than mathematics. A logic that is impoverished and cannot do a "undefined for division by 0 in mathematics". The true logic must be able to have the fact that division by 0 is undefined. True logic is larger than all of mathematics, and must be able to fetch any piece of mathematics from out of Logic itself. So another word for U is undefined. And this is the crux of why Reductio ad Absurdum cannot be a proof method of mathematics, for a starting falsehood in a mathematics proof can only lead to a probability end conclusion.

My corrections of Old Logic have a history that dates before 1993, sometime around 1991, I realized the Euclid proof of infinitude of primes was illogical, sadly sadly wrong, in that the newly formed number by "multiply the lot and add 1" was necessarily a new prime in the indirect proof method. So that my history of fixing Old Logic starts in 1991, but comes to a synthesis of correcting all four of the connectors of Equal/not, And, Or, If->Then, by 2015.

Cover picture: some may complain my covers are less in quality, but I have a good reason for those covers-- I would like covers of math or logic to show the teacher's own handwriting as if he were back in the classroom writing on the blackboard or an overhead projector.
Length: 65 pages

File Size: 764 KB
Print Length: 65 pages
Publication Date: March 12, 2019
Sold by: Amazon Digital Services LLC
Language: English
ASIN: B07PMB69F5
Text-to-Speech: Enabled 
X-Ray: 
Not Enabled  

Word Wise: Not Enabled
Lending: Enabled
Screen Reader: Supported 
Enhanced Typesetting: Enabled 




#6-2, 27th published book

Correcting Reductio Ad Absurdum// Logic series, book 2 Kindle Edition
by Archimedes Plutonium (Author)


These are the TRUE Truth Tables of the 4 connectors of Logic

Equal+Not
T = T = T
T = ~F = T
F = ~T = T
F = F = T

If--> then
T --> T = T
T --> F = F
F --> T = U (unknown or uncertain)
F --> F = U (unknown or uncertain)

And

T & T = T
T & F = T
F & T = T
F & F = F

Or
T or T = F
T or F = T
F or T = T
F or F = F

Those can be analyzed as being Equal+Not is multiplication, If-->then is division, And is addition and Or is subtraction in mathematics. Now I need to emphasis this error of Old Logic, the If->Then conditional. I need to make it clear enough to the reader why the true Truth Table of IF --> Then requires a U for unknown or uncertain with a probability outcome for F --> T = U and F --> F = U. Some smart readers would know that the reason for the U is because without the U, Logic has no means of division by 0 which is undefined in mathematics. You cannot have a Logic that is less than mathematics. A logic that is impoverished and cannot do a "undefined for division by 0 in mathematics". The true logic must be able to have the fact that division by 0 is undefined. True logic is larger than all of mathematics, and must be able to fetch any piece of mathematics from out of Logic itself. So another word for U is undefined. And this is the crux of why Reductio ad Absurdum cannot be a proof method of mathematics, for a starting falsehood in a mathematics proof can only lead to a probability end conclusion.

Now in Old Logic they had for Reductio Ad Absurdum as displayed by this schematic:

| | ~p
| |---
| | .
| | .
| | q
| | .
| | .
| | ~q
| p

Which is fine except for the error of not indicating the end conclusion of "p" is only a probability of being true, not guaranteed as true. And this is the huge huge error that mathematicians have fallen victim of. For the Reductio Ad Absurdum is not a proof method for mathematics, it is probability of being true or false. Math works on guaranteed truth, not probability. This textbook is written to fix that error.

Cover Picture: I like my covers to be like as if a blackboard in school to connect with students. This is a picture of the above Reductio Ad Absurdum, as a student or teacher would write in their notes or blackboard.
Length: 82 pages

File Size: 1175 KB
Print Length: 82 pages
Publication Date: March 23, 2019
Sold by: Amazon Digital Services LLC
Language: English
ASIN: B07Q18GQ7S
Text-to-Speech: Enabled 
X-Ray: 
Not Enabled  

Word Wise: Not Enabled
Lending: Enabled
Enhanced Typesetting: Enabled 

[Dan Christensen]

On Monday, October 19, 2020 at 5:07:54 PM UTC-4, Mostowski Collapse wrote:
> Where?

(Sorry, I don't have time to reformat it. )


THEOREM
*******

P(0,0)
& ALL(b):[b @ n => [P(0,b) => P(0,s(b))]]
& ALL(b):[b @ n => ALL(a):[a @ n => [P(a,b) => P(s(a),b)]]]

=> ALL(a):ALL(b):[a @ n & b @ n => P(a,b)]


PEANO'S AXIOMS
**************

1 Set(n)
Axiom

2 0 @ n
Axiom

3 ALL(a):[a @ n => s(a) @ n]
Axiom

Not used here:

4 ALL(a):ALL(b):[a @ n & b @ n => [s(a)=s(b) => a=b]]
Axiom

Not used here:

5 ALL(a):[a @ n => ~s(a)=0]
Axiom

6 ALL(a):[Set(a) & ALL(b):[b @ a => b @ n]
=> [0 @ a & ALL(b):[b @ a => s(b) @ a]
=> ALL(b):[b @ n => b @ a]]]
Axiom


PROOF
*****

Suppose...

7 P(0,0)
& ALL(b):[b @ n => [P(0,b) => P(0,s(b))]]
& ALL(b):[b @ n => ALL(a):[a @ n => [P(a,b) => P(s(a),b)]]]
Premise

8 P(0,0)
Split, 7

9 ALL(b):[b @ n => [P(0,b) => P(0,s(b))]]
Split, 7

10 ALL(b):[b @ n => ALL(a):[a @ n => [P(a,b) => P(s(a),b)]]]
Split, 7

Construct subset p1 of n

11 EXIST(a):[Set(a) & ALL(b):[b @ a <=> b @ n & P(0,b)]]
Subset, 1

12 Set(p1) & ALL(b):[b @ p1 <=> b @ n & P(0,b)]
E Spec, 11


Define: p1

13 Set(p1)
Split, 12

14 ALL(b):[b @ p1 <=> b @ n & P(0,b)]
Split, 12

Apply Principle of Mathematical Induction

15 Set(p1) & ALL(b):[b @ p1 => b @ n]
=> [0 @ p1 & ALL(b):[b @ p1 => s(b) @ p1]
=> ALL(b):[b @ n => b @ p1]]
U Spec, 6


Prove: ALL(b):[b @ p1 => b @ n]

Suppose...

16 x @ p1
Premise

17 x @ p1 <=> x @ n & P(0,x)
U Spec, 14

18 [x @ p1 => x @ n & P(0,x)] & [x @ n & P(0,x) => x @ p1]
Iff-And, 17

19 x @ p1 => x @ n & P(0,x)
Split, 18

20 x @ n & P(0,x) => x @ p1
Split, 18

21 x @ n & P(0,x)
Detach, 19, 16

22 x @ n
Split, 21

As Required:

23 ALL(b):[b @ p1 => b @ n]
Conclusion, 16

24 Set(p1) & ALL(b):[b @ p1 => b @ n]
Join, 13, 23

25 0 @ p1 & ALL(b):[b @ p1 => s(b) @ p1]
=> ALL(b):[b @ n => b @ p1]
Detach, 15, 24

Base case
*********

Prove: 0 @ p1

26 0 @ p1 <=> 0 @ n & P(0,0)
U Spec, 14

27 [0 @ p1 => 0 @ n & P(0,0)] & [0 @ n & P(0,0) => 0 @ p1]
Iff-And, 26

28 0 @ p1 => 0 @ n & P(0,0)
Split, 27

29 0 @ n & P(0,0) => 0 @ p1
Split, 27

30 0 @ n & P(0,0)
Join, 2, 8

As Required:

31 0 @ p1
Detach, 29, 30


Inductive Step
**************

Prove: ALL(b):[b @ p1 => s(b) @ p1]

Suppose...

32 x @ p1
Premise

33 x @ p1 <=> x @ n & P(0,x)
U Spec, 14

34 [x @ p1 => x @ n & P(0,x)] & [x @ n & P(0,x) => x @ p1]
Iff-And, 33

35 x @ p1 => x @ n & P(0,x)
Split, 34

36 x @ n & P(0,x) => x @ p1
Split, 34

37 x @ n & P(0,x)
Detach, 35, 32

38 x @ n
Split, 37

39 P(0,x)
Split, 37

Prove: s(x) @ p1

40 x @ n => s(x) @ n
U Spec, 3

41 s(x) @ n
Detach, 40, 38

42 s(x) @ p1 <=> s(x) @ n & P(0,s(x))
U Spec, 14, 41

43 [s(x) @ p1 => s(x) @ n & P(0,s(x))]
& [s(x) @ n & P(0,s(x)) => s(x) @ p1]
Iff-And, 42

44 s(x) @ p1 => s(x) @ n & P(0,s(x))
Split, 43

Sufficient: For s(x) @ p1

45 s(x) @ n & P(0,s(x)) => s(x) @ p1
Split, 43

46 x @ n => [P(0,x) => P(0,s(x))]
U Spec, 9

47 P(0,x) => P(0,s(x))
Detach, 46, 38

48 P(0,s(x))
Detach, 47, 39

49 s(x) @ n & P(0,s(x))
Join, 41, 48

As Required:

50 s(x) @ p1
Detach, 45, 49

As Required:

51 ALL(b):[b @ p1 => s(b) @ p1]
Conclusion, 32

52 0 @ p1 & ALL(b):[b @ p1 => s(b) @ p1]
Join, 31, 51

By induction, we have...

53 ALL(b):[b @ n => b @ p1]
Detach, 25, 52


Prove: ALL(b):[b @ n => P(0,b)]

Suppose...

54 x @ n
Premise

55 x @ n => x @ p1
U Spec, 53

56 x @ p1
Detach, 55, 54

57 x @ p1 <=> x @ n & P(0,x)
U Spec, 14

58 [x @ p1 => x @ n & P(0,x)] & [x @ n & P(0,x) => x @ p1]
Iff-And, 57

59 x @ p1 => x @ n & P(0,x)
Split, 58

60 x @ n & P(0,x) => x @ p1
Split, 58

61 x @ n & P(0,x)
Detach, 59, 56

62 x @ n
Split, 61

63 P(0,x)
Split, 61

As Required:

64 ALL(b):[b @ n => P(0,b)]
Conclusion, 54


Prove: ALL(b):[b @ n => ALL(a):[a @ n => P(a,b)]]

Suppose...

65 y @ n
Premise

Construct subset p2 of n

66 EXIST(a):[Set(a) & ALL(b):[b @ a <=> b @ n & P(b,y)]]
Subset, 1

67 Set(p2) & ALL(b):[b @ p2 <=> b @ n & P(b,y)]
E Spec, 66


Define: p2

68 Set(p2)
Split, 67

69 ALL(b):[b @ p2 <=> b @ n & P(b,y)]
Split, 67

Apply PMI

70 Set(p2) & ALL(b):[b @ p2 => b @ n]
=> [0 @ p2 & ALL(b):[b @ p2 => s(b) @ p2]
=> ALL(b):[b @ n => b @ p2]]
U Spec, 6


Prove: ALL(b):[b @ p2 => b @ n]

Suppose...

71 x @ p2
Premise

72 x @ p2 <=> x @ n & P(x,y)
U Spec, 69

73 [x @ p2 => x @ n & P(x,y)] & [x @ n & P(x,y) => x @ p2]
Iff-And, 72

74 x @ p2 => x @ n & P(x,y)
Split, 73

75 x @ n & P(x,y) => x @ p2
Split, 73

76 x @ n & P(x,y)
Detach, 74, 71

77 x @ n
Split, 76

As Required:

78 ALL(b):[b @ p2 => b @ n]
Conclusion, 71

79 Set(p2) & ALL(b):[b @ p2 => b @ n]
Join, 68, 78


Sufficient: For ALL(b):[b @ n => b @ p2]

80 0 @ p2 & ALL(b):[b @ p2 => s(b) @ p2]
=> ALL(b):[b @ n => b @ p2]
Detach, 70, 79


Base case
*********

Prove: 0 @ p2

81 0 @ p2 <=> 0 @ n & P(0,y)
U Spec, 69

82 [0 @ p2 => 0 @ n & P(0,y)] & [0 @ n & P(0,y) => 0 @ p2]
Iff-And, 81

83 0 @ p2 => 0 @ n & P(0,y)
Split, 82

84 0 @ n & P(0,y) => 0 @ p2
Split, 82

85 y @ n => P(0,y)
U Spec, 64

86 P(0,y)
Detach, 85, 65

87 0 @ n & P(0,y)
Join, 2, 86

As Required:

88 0 @ p2
Detach, 84, 87


Inductive step
**************

Prove: ALL(b):[b @ p2 => s(b) @ p2]

Suppose...

89 x @ p2
Premise

90 x @ p2 <=> x @ n & P(x,y)
U Spec, 69

91 [x @ p2 => x @ n & P(x,y)] & [x @ n & P(x,y) => x @ p2]
Iff-And, 90

92 x @ p2 => x @ n & P(x,y)
Split, 91

93 x @ n & P(x,y) => x @ p2
Split, 91

94 x @ n & P(x,y)
Detach, 92, 89

95 x @ n
Split, 94

96 P(x,y)
Split, 94

97 x @ n => s(x) @ n
U Spec, 3

98 s(x) @ n
Detach, 97, 95

99 s(x) @ p2 <=> s(x) @ n & P(s(x),y)
U Spec, 69, 98

100 [s(x) @ p2 => s(x) @ n & P(s(x),y)]
& [s(x) @ n & P(s(x),y) => s(x) @ p2]
Iff-And, 99

101 s(x) @ p2 => s(x) @ n & P(s(x),y)
Split, 100

Sufficient: For s(x) @ p2

102 s(x) @ n & P(s(x),y) => s(x) @ p2
Split, 100

103 y @ n => ALL(a):[a @ n => [P(a,y) => P(s(a),y)]]
U Spec, 10

104 ALL(a):[a @ n => [P(a,y) => P(s(a),y)]]
Detach, 103, 65

105 x @ n => [P(x,y) => P(s(x),y)]
U Spec, 104

106 P(x,y) => P(s(x),y)
Detach, 105, 95

107 P(s(x),y)
Detach, 106, 96

108 s(x) @ n & P(s(x),y)
Join, 98, 107

109 s(x) @ p2
Detach, 102, 108

As Required:

110 ALL(b):[b @ p2 => s(b) @ p2]
Conclusion, 89

111 0 @ p2 & ALL(b):[b @ p2 => s(b) @ p2]
Join, 88, 110

By induction, we have...

112 ALL(b):[b @ n => b @ p2]
Detach, 80, 111


Prove: ALL(a):[a @ n => P(a,y)]

Suppose...

113 x @ n
Premise

114 x @ n => x @ p2
U Spec, 112

115 x @ p2
Detach, 114, 113

116 x @ p2 <=> x @ n & P(x,y)
U Spec, 69

117 [x @ p2 => x @ n & P(x,y)] & [x @ n & P(x,y) => x @ p2]
Iff-And, 116

118 x @ p2 => x @ n & P(x,y)
Split, 117

119 x @ n & P(x,y) => x @ p2
Split, 117

120 x @ n & P(x,y)
Detach, 118, 115

121 x @ n
Split, 120

122 P(x,y)
Split, 120

As Required:

123 ALL(a):[a @ n => P(a,y)]
Conclusion, 113

As Required:

124 ALL(b):[b @ n => ALL(a):[a @ n => P(a,b)]]
Conclusion, 65


Prove: ALL(a):ALL(b):[a @ n & b @ n => P(a,b)]

Suppose...

125 x @ n & y @ n
Premise

126 x @ n
Split, 125

127 y @ n
Split, 125

128 y @ n => ALL(a):[a @ n => P(a,y)]
U Spec, 124

129 ALL(a):[a @ n => P(a,y)]
Detach, 128, 127

130 x @ n => P(x,y)
U Spec, 129

131 P(x,y)
Detach, 130, 126

As Required:

132 ALL(a):ALL(b):[a @ n & b @ n => P(a,b)]
Conclusion, 125

As Required:

133 P(0,0)
& ALL(b):[b @ n => [P(0,b) => P(0,s(b))]]
& ALL(b):[b @ n => ALL(a):[a @ n => [P(a,b) => P(s(a),b)]]]
=> ALL(a):ALL(b):[a @ n & b @ n => P(a,b)]
Conclusion, 7

[Mostowski Collapse]

Can we extract a well founded relation from
your induction schema. Lets say we define
a one step relation:

/* Dan-O-Matic ordering */

<0,b> <_1 <0,b'>

<a,b> <_1 <a',b>

What is the transitive closure of <_1 ? Is
it the same as the omega^2 ordering? Can
the omega^2 ordering have a one step relation,

i.e. does it have direct predecessors? Actually
possibly no, since for omega^2 we would have:

/* omega^2 ordering */

<a,b> <_1 <a,b'>

<a,c> <_1 <a',b>

c can be arbitrary, even larger than b. Thats
the point of primary / secondary key in lexical
sorting.

So the order type of Dan-O-Matic induction is
not omega^2, what is it?

On Tuesday, October 20, 2020 at 12:13:07 AM UTC+2, Dan Christensen wrote:
> On Monday, October 19, 2020 at 5:07:54 PM UTC-4, Mostowski Collapse wrote:
> > Where?
>
> (Sorry, I don't have time to reformat it. )

[Mostowski Collapse]

Possibly the Dan-O-Matic ordering is not
linear. It is neither:

<3,4> <_1 <4,5>

Nor:

<4,5> <_1 <3,4>

Also not transitively?

[Graham Cooper]

On Tuesday, October 20, 2020 at 1:05:26 AM UTC+10, Dan Christensen wrote:
> Recall: PROOF BY INDUCTION FOR 1 VARIABLE
>
> Suppose you want to prove that, for all m in N, we have P(m) being true.
>
> It will be sufficient to prove:
>
> 1. P(0)
>
> 2. For all k in N, we have: P(k) => P(k+1)
>
>
> *****************************************************************************
>
> PROOF BY INDUCTION FOR 2 VARIABLES
>
> Suppose you want to prove that, for all m, n in N, we have P(m, n) being true.
>
> It will be sufficient to prove:
>
> 1. P(0,0)
>
> 2. For all k in N, we have: P(0, k) => P(0, k+1)
>
> 3. For all j, k in N, we have: P(j, k) => P(j+1, k)
>
>
> Your comments?
>

a simple example

prove a+b = b+a

0+b=b+0 -> 0+b+1=b+1+0
CHECK

a+b=b+a -> a+1+b=b+a+1
CHECK


Not seeing it...

How about

P(0 0)
P(0 b) -> P(0 b+1)
P(a 0) -> P(a+1 0)
P(a b) -> P(a+1 b)
P(a b) -> P(a b+1)

[Graham Cooper]

this is overkill..

(00)
**********
* v
* v
* > > >
*
*


So you get

P( 0 0 )
P( 0 y ) -> P( 0 y+1 ) *
P( x y ) -> P( x+1 y ) >

* >>>
* >>>
* >>>



OR

P( 0 0 )
P( x 0 ) -> P( x+1 0 ) *
P( x y ) -> P( x y+1 ) v

****
vvv
vvv
vvv



----------

IE

P(0 0)

P( 0 y ) -> P( 0 y+1 )
P( x y ) -> P( x+1 y )
OR
P( x 0 ) -> P( x+1 0 )
P( x y ) -> P( x y+1 )



again Dan was half - right!

[Brody Roccanello]

Mostowski Collapse wrote:

> Possibly the Dan-O-Matic ordering is not linear. It is neither:
> <3,4> <_1 <4,5> Nor: <4,5> <_1 <3,4> Also not transitively?

the stupidest lines I ever see. But your *covid-19* is no longer a
pandemic, but a *_syndemic_*, which is about obesity, sedentary and
climate change. No joke, when you move around. What can you say in your
defence.

https://www.marca.com/en/lifestyle/
2020/10/15/5f87777c268e3ec9778b45ac.html

https://www.thelancet.com/journals/lancet/article/
PIIS0140-6736(20)32000-6/fulltext

[Dan Christensen]

On Tuesday, October 20, 2020 at 8:29:28 AM UTC-4, Graham Cooper wrote:
> On Tuesday, October 20, 2020 at 1:05:26 AM UTC+10, Dan Christensen wrote:
> > Recall: PROOF BY INDUCTION FOR 1 VARIABLE
> >
> > Suppose you want to prove that, for all m in N, we have P(m) being true.
> >
> > It will be sufficient to prove:
> >
> > 1. P(0)
> >
> > 2. For all k in N, we have: P(k) => P(k+1)
> >
> >
> > *****************************************************************************
> >
> > PROOF BY INDUCTION FOR 2 VARIABLES
> >
> > Suppose you want to prove that, for all m, n in N, we have P(m, n) being true.
> >
> > It will be sufficient to prove:
> >
> > 1. P(0,0)
> >
> > 2. For all k in N, we have: P(0, k) => P(0, k+1)
> >
> > 3. For all j, k in N, we have: P(j, k) => P(j+1, k)
> >
> >
> > Your comments?
> >
>
>
> a simple example
>
> prove a+b = b+a
>

Not so easy. IIRC to prove the commutativity of + on N, you must first prove cancelability (a+b=c+b => a=c), and then associativity ((a+b)+c=a+(b+c)). Both require multiple proofs by induction.

[Graham Cooper]

Maybe assume cancellability and associativity.

What about TRANSPOSE INDUCTION ?


p(0 0)
p(0 y) -> p(y+1 y+1)
x>0 -> p(x y) -> p(x-1 y+1)

*

==>

*
*

==>

**
*

==>

**
*
*

and so on

[Dan Christensen]

On Monday, October 19, 2020 at 11:05:26 AM UTC-4, Dan Christensen wrote:

> PROOF BY INDUCTION FOR 2 VARIABLES
>
> Suppose you want to prove that, for all m, n in N, we have P(m, n) being true.
>
> It will be sufficient to prove:
>
> 1. P(0,0)
>
> 2. For all k in N, we have: P(0, k) => P(0, k+1)
>
> 3. For all j, k in N, we have: P(j, k) => P(j+1, k)
>
>

And now...

Proof by Induction with *** THREE *** Variables


Suppose you want to prove that P(x, y, z) is true for all x, y, z in N. It will suffice to prove each of the following?

1. P(0, 0, 0)

2. For all k in N:[P(0, 0, k) => P(0, 0, k+1)]

3. For all j, k in N:[P(0, j, k) => P(0, j+1, k)]

4. For all i, j, k in N:[P(i,j,k) => P(i+1,j ,k)

[Dan Christensen]

On Tuesday, October 20, 2020 at 11:24:05 PM UTC-4, Archimedes Plutonium wrote:
> Archimedes Plutonium, mindless shithead of logic and math...

WARNING TO STUDENTS: Don't be a victim of AP's fake math and science

AP is a malicious troll who really, REALLY wants you to fail in school just like he must have so long ago (in the 60's?). Then he would like to recruit you to his sinister Atom God Cult of Failure. Think I'm making this up? IN HIS OWN WORDS:


AP's fake math that can only be designed to promote failure in schools:

“Primes do not exist, because the set they were borne from has no division.”
--June 29, 2020

“The last and largest finite number is 10^604.”
--June 3, 2015

“0 appears to be the last and largest finite number”
--June 9, 2015

“0/0 must be equal to 1.”
-- June 9, 2015

“0 is an infinite irrational number.”
--June 28, 2015

“No negative numbers exist.”
--December 22, 2018

“Rationals are not numbers.”
--May 18, 2019

“The value of sin(45 degrees) = 1.” (Actually, sin(45 degrees) = 0.707. tan(45 degrees) = 1.)
--May 31, 2019

AP deliberately and repeatedly presented the truth table for OR as the truth table for AND:

“New Logic
AND

T & T = T
T & F = T
F & T = T

F & F = F”
--November 9, 2019

According to AP's “chess board math,” an equilateral triangle is right-triangle.
--December 11, 2019


AP seeks aid of Russian agents to promote failure in schools:

"Please--Asking for help from Russia-- russian robots-- to create a new, true mathematics [sic]"
--November 9, 2017


And if that wasn't weird enough...


AP's sinister Atom God Cult

“The totality, everything that there is [the universe], is only 1 atom of plutonium [Pu]. There is nothing outside or beyond this one atom of plutonium.”
--April 4, 1994

“The Universe itself is one gigantic big atom.”
--November 14, 2019

“Since God-Pu is marching on.
Glory! Glory! Atom Plutonium!
Its truth is marching on.
It has sounded forth the trumpet that shall never call retreat;
It is sifting out the hearts of people before its judgment seat;
Oh, be swift, my soul, to answer it; be jubilant, my feet!
Our God-Pu is marching on.”
--December 15, 2018 (Note: Pu is the atomic symbol for plutonium)

[Dan Christensen]

On Tuesday, October 20, 2020 at 5:54:13 PM UTC-4, Dan Christensen wrote:
> On Monday, October 19, 2020 at 11:05:26 AM UTC-4, Dan Christensen wrote:
>
> > PROOF BY INDUCTION FOR 2 VARIABLES
> >
> > Suppose you want to prove that, for all m, n in N, we have P(m, n) being true.
> >
> > It will be sufficient to prove:
> >
> > 1. P(0,0)
> >
> > 2. For all k in N, we have: P(0, k) => P(0, k+1)
> >
> > 3. For all j, k in N, we have: P(j, k) => P(j+1, k)
> >
> >
>
> And now...
>
> Proof by Induction with *** THREE *** Variables
>
>
> Suppose you want to prove that P(x, y, z) is true for all x, y, z in N. It will suffice to prove each of the following?
>
> 1. P(0, 0, 0)
>
> 2. For all k in N:[P(0, 0, k) => P(0, 0, k+1)]
>
> 3. For all j, k in N:[P(0, j, k) => P(0, j+1, k)]
>
> 4. For all i, j, k in N:[P(i,j,k) => P(i+1,j ,k)
>
>

I used sets of ordered triples instead of logical predicates in the following proof (205 lines).


THEOREM

ALL(a):[Set''(a) & ALL(b):ALL(c):ALL(d):[(b,c,d) in a => b in n & c in n & d in n]
=> [(0,0,0) in a
& ALL(d):[d in n => [(0,0,d) in a => (0,0,s(d)) in a]]
& ALL(c):ALL(d):[c in n & d in n => [(0,c,d) in a => (0,s(c),d) in a]]
& ALL(b):ALL(c):ALL(d):[b in n & c in n & d in n => [(b,c,d) in a => (s(b),c,d) in a]]

=> ALL(b):ALL(c):ALL(d):[b in n & c in n & d in n => (b,c,d) in a]]]


PEANO'S AXIOMS

1 Set(n)
Axiom

2 0 in n
Axiom

3 ALL(a):[a in n => s(a) in n]
Axiom

Not used here:

4 ALL(a):ALL(b):[a in n & b in n => [s(a)=s(b) => a=b]]
Axiom

Not used here:

5 ALL(a):[a in n => ~s(a)=0]
Axiom

6 ALL(a):[Set(a) & ALL(b):[b in a => b in n]
=> [0 in a & ALL(b):[b in a => s(b) in a]
=> ALL(b):[b in n => b in a]]]
Axiom


Suppose...

7 Set''(p) & ALL(b):ALL(c):ALL(d):[(b,c,d) in p => b in n & c in n & d in n]
Premise


Suppose...

8 (0,0,0) in p
& ALL(d):[d in n => [(0,0,d) in p => (0,0,s(d)) in p]]
& ALL(c):ALL(d):[c in n & d in n => [(0,c,d) in p => (0,s(c),d) in p]]
& ALL(b):ALL(c):ALL(d):[b in n & c in n & d in n => [(b,c,d) in p => (s(b),c,d) in p]]
Premise

9 (0,0,0) in p
Split, 8

10 ALL(d):[d in n => [(0,0,d) in p => (0,0,s(d)) in p]]
Split, 8

11 ALL(c):ALL(d):[c in n & d in n => [(0,c,d) in p => (0,s(c),d) in p]]
Split, 8

12 ALL(b):ALL(c):ALL(d):[b in n & c in n & d in n => [(b,c,d) in p => (s(b),c,d) in p]]
Split, 8

Construct subset p1 of n

13 EXIST(a):[Set(a) & ALL(b):[b in a <=> b in n & (0,0,b) in p]]
Subset, 1

14 Set(p1) & ALL(b):[b in p1 <=> b in n & (0,0,b) in p]
E Spec, 13


Define: p1

15 Set(p1)
Split, 14

16 ALL(b):[b in p1 <=> b in n & (0,0,b) in p]
Split, 14


Apply PMI

17 Set(p1) & ALL(b):[b in p1 => b in n]
=> [0 in p1 & ALL(b):[b in p1 => s(b) in p1]
=> ALL(b):[b in n => b in p1]]
U Spec, 6


Prove: ALL(b):[b in p1 => b in n]

Suppose...

18 x in p1
Premise

19 x in p1 <=> x in n & (0,0,x) in p
U Spec, 16

20 [x in p1 => x in n & (0,0,x) in p]
& [x in n & (0,0,x) in p => x in p1]
Iff-And, 19

21 x in p1 => x in n & (0,0,x) in p
Split, 20

22 x in n & (0,0,x) in p => x in p1
Split, 20

23 x in n & (0,0,x) in p
Detach, 21, 18

24 x in n
Split, 23

As Required:

25 ALL(b):[b in p1 => b in n]
Conclusion, 18

26 Set(p1) & ALL(b):[b in p1 => b in n]
Join, 15, 25


Sufficient: For ALL(b):[b in n => b in p1]

27 0 in p1 & ALL(b):[b in p1 => s(b) in p1]
=> ALL(b):[b in n => b in p1]
Detach, 17, 26


BASE CASE

28 0 in p1 <=> 0 in n & (0,0,0) in p
U Spec, 16

29 [0 in p1 => 0 in n & (0,0,0) in p]
& [0 in n & (0,0,0) in p => 0 in p1]
Iff-And, 28

30 0 in p1 => 0 in n & (0,0,0) in p
Split, 29

31 0 in n & (0,0,0) in p => 0 in p1
Split, 29

32 0 in n & (0,0,0) in p
Join, 2, 9

As Required:

33 0 in p1
Detach, 31, 32


INDUCTIVE STEP

Prove: ALL(b):[b in p1 => s(b) in p1]

Suppose...

34 x in p1
Premise

35 x in p1 <=> x in n & (0,0,x) in p
U Spec, 16

36 [x in p1 => x in n & (0,0,x) in p]
& [x in n & (0,0,x) in p => x in p1]
Iff-And, 35

37 x in p1 => x in n & (0,0,x) in p
Split, 36

38 x in n & (0,0,x) in p => x in p1
Split, 36

39 x in n & (0,0,x) in p
Detach, 37, 34

40 x in n
Split, 39

41 (0,0,x) in p
Split, 39

42 x in n => s(x) in n
U Spec, 3

43 s(x) in n
Detach, 42, 40

44 s(x) in p1 <=> s(x) in n & (0,0,s(x)) in p
U Spec, 16, 43

45 [s(x) in p1 => s(x) in n & (0,0,s(x)) in p]
& [s(x) in n & (0,0,s(x)) in p => s(x) in p1]
Iff-And, 44

46 s(x) in p1 => s(x) in n & (0,0,s(x)) in p
Split, 45

Sufficient: For s(x) in p1

47 s(x) in n & (0,0,s(x)) in p => s(x) in p1
Split, 45

48 x in n => [(0,0,x) in p => (0,0,s(x)) in p]
U Spec, 10

49 (0,0,x) in p => (0,0,s(x)) in p
Detach, 48, 40

50 (0,0,s(x)) in p
Detach, 49, 41

51 s(x) in n & (0,0,s(x)) in p
Join, 43, 50

52 s(x) in p1
Detach, 47, 51

As Required:

53 ALL(b):[b in p1 => s(b) in p1]
Conclusion, 34

54 0 in p1 & ALL(b):[b in p1 => s(b) in p1]
Join, 33, 53

As Required:

55 ALL(b):[b in n => b in p1]
Detach, 27, 54


Prove: ALL(c):[c in n => (0,0,c) in p]

Suppose...

56 x in n
Premise

57 x in n => x in p1
U Spec, 55

58 x in p1
Detach, 57, 56

59 x in p1 <=> x in n & (0,0,x) in p
U Spec, 16

60 [x in p1 => x in n & (0,0,x) in p]
& [x in n & (0,0,x) in p => x in p1]
Iff-And, 59

61 x in p1 => x in n & (0,0,x) in p
Split, 60

62 x in n & (0,0,x) in p => x in p1
Split, 60

63 x in n & (0,0,x) in p
Detach, 61, 58

64 x in n
Split, 63

65 (0,0,x) in p
Split, 63

As Required:

66 ALL(c):[c in n => (0,0,c) in p]
Conclusion, 56


Suppose...

67 z in n

Premise

Construct subset p2 of n

68 EXIST(a):[Set(a) & ALL(b):[b in a <=> b in n & (0,b,z) in p]]
Subset, 1

69 Set(p2) & ALL(b):[b in p2 <=> b in n & (0,b,z) in p]
E Spec, 68

Define: p2

70 Set(p2)
Split, 69

71 ALL(b):[b in p2 <=> b in n & (0,b,z) in p]
Split, 69

Apply PMI

72 Set(p2) & ALL(b):[b in p2 => b in n]
=> [0 in p2 & ALL(b):[b in p2 => s(b) in p2]
=> ALL(b):[b in n => b in p2]]
U Spec, 6


Prove: ALL(b):[b in p2 => b in n]

Suppose...

73 x in p2
Premise

74 x in p2 <=> x in n & (0,x,z) in p
U Spec, 71

75 [x in p2 => x in n & (0,x,z) in p]
& [x in n & (0,x,z) in p => x in p2]
Iff-And, 74

76 x in p2 => x in n & (0,x,z) in p
Split, 75

77 x in n & (0,x,z) in p => x in p2
Split, 75

78 x in n & (0,x,z) in p
Detach, 76, 73

79 x in n
Split, 78

As Required:

80 ALL(b):[b in p2 => b in n]
Conclusion, 73

81 Set(p2) & ALL(b):[b in p2 => b in n]
Join, 70, 80

Sufficient: For ALL(b):[b in n => b in p2]

82 0 in p2 & ALL(b):[b in p2 => s(b) in p2]
=> ALL(b):[b in n => b in p2]
Detach, 72, 81


BASE CASE

83 0 in p2 <=> 0 in n & (0,0,z) in p
U Spec, 71

84 [0 in p2 => 0 in n & (0,0,z) in p]
& [0 in n & (0,0,z) in p => 0 in p2]
Iff-And, 83

85 0 in p2 => 0 in n & (0,0,z) in p
Split, 84

Sufficient: For 0 in p2

86 0 in n & (0,0,z) in p => 0 in p2
Split, 84

87 z in n => (0,0,z) in p
U Spec, 66

88 (0,0,z) in p
Detach, 87, 67

89 0 in n & (0,0,z) in p
Join, 2, 88

As Required:

90 0 in p2
Detach, 86, 89


INDUCTIVE STEP

Suppose...

91 y in p2
Premise

92 y in p2 <=> y in n & (0,y,z) in p
U Spec, 71

93 [y in p2 => y in n & (0,y,z) in p]
& [y in n & (0,y,z) in p => y in p2]
Iff-And, 92

94 y in p2 => y in n & (0,y,z) in p
Split, 93

95 y in n & (0,y,z) in p => y in p2
Split, 93

96 y in n & (0,y,z) in p
Detach, 94, 91

97 y in n
Split, 96

98 (0,y,z) in p
Split, 96

99 y in n => s(y) in n
U Spec, 3

100 s(y) in n
Detach, 99, 97

101 s(y) in p2 <=> s(y) in n & (0,s(y),z) in p
U Spec, 71, 100

102 [s(y) in p2 => s(y) in n & (0,s(y),z) in p]
& [s(y) in n & (0,s(y),z) in p => s(y) in p2]
Iff-And, 101

103 s(y) in p2 => s(y) in n & (0,s(y),z) in p
Split, 102

104 s(y) in n & (0,s(y),z) in p => s(y) in p2
Split, 102

105 ALL(d):[y in n & d in n => [(0,y,d) in p => (0,s(y),d) in p]]
U Spec, 11

106 y in n & z in n => [(0,y,z) in p => (0,s(y),z) in p]
U Spec, 105

107 y in n & z in n
Join, 97, 67

108 (0,y,z) in p => (0,s(y),z) in p
Detach, 106, 107

109 (0,s(y),z) in p
Detach, 108, 98

110 s(y) in n & (0,s(y),z) in p
Join, 100, 109

111 s(y) in p2
Detach, 104, 110

As Required:

112 ALL(b):[b in p2 => s(b) in p2]
Conclusion, 91

113 0 in p2 & ALL(b):[b in p2 => s(b) in p2]
Join, 90, 112

As Required:

114 ALL(b):[b in n => b in p2]
Detach, 82, 113


Prove: ALL(b):[b in n => (0,b,z) in p]

Suppose...

115 y in n
Premise

116 y in n => y in p2
U Spec, 114

117 y in p2
Detach, 116, 115

118 y in p2 <=> y in n & (0,y,z) in p
U Spec, 71

119 [y in p2 => y in n & (0,y,z) in p]
& [y in n & (0,y,z) in p => y in p2]
Iff-And, 118

120 y in p2 => y in n & (0,y,z) in p
Split, 119

121 y in n & (0,y,z) in p => y in p2
Split, 119

122 y in n & (0,y,z) in p
Detach, 120, 117

123 y in n
Split, 122

124 (0,y,z) in p
Split, 122

As Required:

125 ALL(b):[b in n => (0,b,z) in p]
Conclusion, 115

As Required:

126 ALL(c):[c in n => ALL(b):[b in n => (0,b,c) in p]]
Conclusion, 67


Prove: ALL(b):ALL(c):[b in n & c in n => ALL(a):[a in n => (a,b,c) in p]]

Suppose...

127 y in n & z in n
Premise

128 y in n
Split, 127

129 z in n
Split, 127

130 EXIST(a):[Set(a) & ALL(b):[b in a <=> b in n & (b,y,z) in p]]
Subset, 1

131 Set(p3) & ALL(b):[b in p3 <=> b in n & (b,y,z) in p]
E Spec, 130


Define: p3

132 Set(p3)
Split, 131

133 ALL(b):[b in p3 <=> b in n & (b,y,z) in p]
Split, 131

Apply PMI

134 Set(p3) & ALL(b):[b in p3 => b in n]
=> [0 in p3 & ALL(b):[b in p3 => s(b) in p3]
=> ALL(b):[b in n => b in p3]]
U Spec, 6


Prove: ALL(b):[b in p3 => b in n]

Suppose...

135 x in p3
Premise

136 x in p3 <=> x in n & (x,y,z) in p
U Spec, 133

137 [x in p3 => x in n & (x,y,z) in p]
& [x in n & (x,y,z) in p => x in p3]
Iff-And, 136

138 x in p3 => x in n & (x,y,z) in p
Split, 137

139 x in n & (x,y,z) in p => x in p3
Split, 137

140 x in n & (x,y,z) in p
Detach, 138, 135

141 x in n
Split, 140

As Required:

142 ALL(b):[b in p3 => b in n]
Conclusion, 135

143 Set(p3) & ALL(b):[b in p3 => b in n]
Join, 132, 142


Sufficient: For ALL(b):[b in n => b in p3]

144 0 in p3 & ALL(b):[b in p3 => s(b) in p3]
=> ALL(b):[b in n => b in p3]
Detach, 134, 143


BASE CASE

145 0 in p3 <=> 0 in n & (0,y,z) in p
U Spec, 133

146 [0 in p3 => 0 in n & (0,y,z) in p]
& [0 in n & (0,y,z) in p => 0 in p3]
Iff-And, 145

147 0 in p3 => 0 in n & (0,y,z) in p
Split, 146

148 0 in n & (0,y,z) in p => 0 in p3
Split, 146

149 z in n => ALL(b):[b in n => (0,b,z) in p]
U Spec, 126

150 ALL(b):[b in n => (0,b,z) in p]
Detach, 149, 129

151 y in n => (0,y,z) in p
U Spec, 150

152 (0,y,z) in p
Detach, 151, 128

153 0 in n & (0,y,z) in p
Join, 2, 152

As Required:

154 0 in p3
Detach, 148, 153


INDUCTIVE STEP

Prove: ALL(a):[a in p3 => s(a) in p3]

Suppose...

155 x in p3
Premise

156 x in p3 <=> x in n & (x,y,z) in p
U Spec, 133

157 [x in p3 => x in n & (x,y,z) in p]
& [x in n & (x,y,z) in p => x in p3]
Iff-And, 156

158 x in p3 => x in n & (x,y,z) in p
Split, 157

159 x in n & (x,y,z) in p => x in p3
Split, 157

160 x in n & (x,y,z) in p
Detach, 158, 155

161 x in n
Split, 160

162 (x,y,z) in p
Split, 160

163 x in n => s(x) in n
U Spec, 3

164 s(x) in n
Detach, 163, 161

165 s(x) in p3 <=> s(x) in n & (s(x),y,z) in p
U Spec, 133, 164

166 [s(x) in p3 => s(x) in n & (s(x),y,z) in p]
& [s(x) in n & (s(x),y,z) in p => s(x) in p3]
Iff-And, 165

167 s(x) in p3 => s(x) in n & (s(x),y,z) in p
Split, 166

168 s(x) in n & (s(x),y,z) in p => s(x) in p3
Split, 166

169 ALL(c):ALL(d):[x in n & c in n & d in n => [(x,c,d) in p => (s(x),c,d) in p]]
U Spec, 12

170 ALL(d):[x in n & y in n & d in n => [(x,y,d) in p => (s(x),y,d) in p]]
U Spec, 169

171 x in n & y in n & z in n => [(x,y,z) in p => (s(x),y,z) in p]
U Spec, 170

172 x in n & y in n
Join, 161, 128

173 x in n & y in n & z in n
Join, 172, 129

174 (x,y,z) in p => (s(x),y,z) in p
Detach, 171, 173

175 (s(x),y,z) in p
Detach, 174, 162

176 s(x) in n & (s(x),y,z) in p
Join, 164, 175

177 s(x) in p3
Detach, 168, 176

As Required:

178 ALL(a):[a in p3 => s(a) in p3]
Conclusion, 155

179 0 in p3 & ALL(a):[a in p3 => s(a) in p3]
Join, 154, 178

180 ALL(b):[b in n => b in p3]
Detach, 144, 179


Prove: ALL(a):[a in n => (a,y,z) in p]

Suppose...

181 x in n
Premise

182 x in n => x in p3
U Spec, 180

183 x in p3
Detach, 182, 181

184 x in p3 <=> x in n & (x,y,z) in p
U Spec, 133

185 [x in p3 => x in n & (x,y,z) in p]
& [x in n & (x,y,z) in p => x in p3]
Iff-And, 184

186 x in p3 => x in n & (x,y,z) in p
Split, 185

187 x in n & (x,y,z) in p => x in p3
Split, 185

188 x in n & (x,y,z) in p
Detach, 186, 183

189 x in n
Split, 188

190 (x,y,z) in p
Split, 188

As Required:

191 ALL(a):[a in n => (a,y,z) in p]
Conclusion, 181

As Required:

192 ALL(b):ALL(c):[b in n & c in n => ALL(a):[a in n => (a,b,c) in p]]
Conclusion, 127


Prove: ALL(a):ALL(b):ALL(c):[a in n & b in n & c in n => (a,b,c) in p]

Suppose...

193 x in n & y in n & z in n
Premise

194 x in n
Split, 193

195 y in n
Split, 193

196 z in n
Split, 193

197 ALL(c):[y in n & c in n => ALL(a):[a in n => (a,y,c) in p]]
U Spec, 192

198 y in n & z in n => ALL(a):[a in n => (a,y,z) in p]
U Spec, 197

199 y in n & z in n
Join, 195, 196

200 ALL(a):[a in n => (a,y,z) in p]
Detach, 198, 199

201 x in n => (x,y,z) in p
U Spec, 200

202 (x,y,z) in p
Detach, 201, 194

As Required:

203 ALL(a):ALL(b):ALL(c):[a in n & b in n & c in n => (a,b,c) in p]
Conclusion, 193

As Required:

204 (0,0,0) in p
& ALL(d):[d in n => [(0,0,d) in p => (0,0,s(d)) in p]]
& ALL(c):ALL(d):[c in n & d in n => [(0,c,d) in p => (0,s(c),d) in p]]
& ALL(b):ALL(c):ALL(d):[b in n & c in n & d in n => [(b,c,d) in p => (s(b),c,d) in p]]
=> ALL(a):ALL(b):ALL(c):[a in n & b in n & c in n => (a,b,c) in p]
Conclusion, 8

As Required:

205 ALL(p):[Set''(p) & ALL(b):ALL(c):ALL(d):[(b,c,d) in p => b in n & c in n & d in n]
=> [(0,0,0) in p
& ALL(d):[d in n => [(0,0,d) in p => (0,0,s(d)) in p]]
& ALL(c):ALL(d):[c in n & d in n => [(0,c,d) in p => (0,s(c),d) in p]]
& ALL(b):ALL(c):ALL(d):[b in n & c in n & d in n => [(b,c,d) in p => (s(b),c,d) in p]]
=> ALL(a):ALL(b):ALL(c):[a in n & b in n & c in n => (a,b,c) in p]]]
Conclusion, 7

[Archimedes Plutonium]

Dan Christensen & Jan Burse, mindless shitheads of logic and math with their 10 OR 2=12 and their 10 AND 2 as subtraction, they are such mindless turds of math and logic they never have a geometry proof of Calculus and they think ellipse is a conic


Learn the 50 fakes and mistakes of Old Math, especially no geometry proof of Fundamental Theorem of Calculus by AP's TEACHING TRUE MATHEMATICS

134th published book

Introduction to TEACHING TRUE MATHEMATICS: Volume 1 for ages 5 through 26, math textbook series, book 1 Kindle Edition
by Archimedes Plutonium (Author)

The 134th book of AP, and belatedly late, for I had already written the series of TEACHING TRUE MATHEMATICS in a 7 volume, 8 book set. This would be the first book in that 8 book set (one of the books is a companion book to 1st year college). But I suppose that I needed to write the full series before I could write the Introduction and know what I had to talk about and talk about in a logical progression order. Sounds paradoxical in a sense, that I needed to write the full series first and then go back and write the Introduction. But in another sense, hard to write an introduction on something you have not really fully done and completed. For example to know what is error filled Old Math and to list those errors in a logical order requires me to write the full 7 volumes in order to list in order the mistakes.

Cover Picture: Mathematics begins with counting, with numbers, with quantity. But counting numbers needs geometry for something to count in the first place. So here in this picture of the generalized Hydrogen atom of chemistry and physics is a torus geometry of 8 rings of a proton torus and one ring where my fingers are, is a equator ring that is the muon and thrusting through the proton torus at the equator of the torus. So we count 9 rings in all. So math is created by atoms and math numbers exist because atoms have many geometry figures to count. And geometry exists because atoms have shapes and different figures.

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#4-2, 45th published book

TEACHING TRUE MATHEMATICS: Volume 2 for ages 5 to 18, math textbook series, book 2 Kindle Edition
by Archimedes Plutonium (Author)




#1 New Releasein General Geometry


Volume 1 takes the 5 year old student through to senior in High School for their math education.

This is the one textbook series in several volumes that carries every person through all his/her math education starting age 5 up to age 26. Volume 1 is for age 5 year old to that of senior in High School, that is needed to do both science and math. Every other math book is incidental to this series of Teaching True Mathematics.

I call it a journal-textbook because Amazon's Kindle offers me the ability to edit overnight, and to change the text, almost on a daily basis. A unique first in education textbooks-- almost a continual overnight editing. Adding new text, correcting text. Volume 1 takes the 5 year old student through to senior in High School for their math education. Volume 2 carries the Freshperson in College for their math calculus education.

Cover Picture: The Numbers as Integers from 0 to 100, and 10 Grid when dividing by 10, and part of the 100 Grid when dividing by 100. Decimal Grid Numbers are the true numbers of mathematics. The Reals, the rationals & irrationals, the algebraic & transcendentals, the imaginary & Complex, and the negative-numbers are all fake numbers. For, to be a true number, you have to "be counted" by mathematical induction. The smallest Grid system is the Decimal 10 Grid.

Length: 375 pages






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Amazon Best Sellers Rank: #274,398 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
#27 in General Geometry
#336 in Geometry & Topology (Books)




#4-3, 55th published book

TEACHING TRUE MATHEMATICS: Volume 3 for age 18-19, 1st year College Calculus, math textbook series, book 3 Kindle Edition
by Archimedes Plutonium (Author)


This is volume 2, series 2 of Teaching True Mathematics, designed for College Freshperson of age 18-19. This is 1st year college calculus math. It is the continuation of volume 1 for ages 5 through 18 years old. The main major topic is the AP-EM equations of electricity and magnetism. The end goal of this textbook is to teach all math, physics, chemistry, biology and hard sciences the mathematics they need to continue in science. This textbook at the end teaches students the mathematics of electricity and magnetism. In Old Math, they devoted about 1/3 of their textbook content to reach the Maxwell Equations at the end, but sadly, no Old Math textbook was ever able to cover the Maxwell Equations with any comprehension, and left the students with headaches, nightmares and nervous breakdowns of the mathematics needed to do electricity and magnetism. This textbook resolves all those problems. And this textbook asks the students and parents of students to equip their daughter or son with a tool kit of instruments to do electricity and magnetism.
Cover Picture is my EM instruments to gain numbers for the laws of EM theory: a multimeter, galvanometer attached to a coil with a strong bar magnet inside. Three electromagnets and a transformer to step down 120 volts to 12 volts to run the electromagnets. A case of iron powder to measure the strength of magnets and a battery case. I urge students who can afford these instruments to use them as you take this course in calculus.
Length: 140 pages

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Publication Date: August 16, 2019
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Amazon Best Sellers Rank: #1,212,707 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
#2719 in Calculus (Books)
#417 in Calculus (Kindle Store)



#4-4, 56th published book

COLLEGE CALCULUS GUIDE to help students recognize math professor spam from math truth & reality// math textbook series, book 4 Kindle Edition

by Archimedes Plutonium (Author)


#1 New Releasein 15-Minute Science & Math Short Reads


This textbook is the companion guide book to AP's Teaching True Mathematics, 1st year College. It is realized that Old Math will take a long time in removing their fake math, so in the interim period, this Guide book is designed to speed up the process of removing fake Calculus out of the education system, the fewer students we punish with forcing them with fake Calculus, the better we are.
Cover Picture: This book is part comedy, for when you cannot reason with math professors that they have many errors to fix, that 90% of their Calculus is in error, you end up resorting to comedy, making fun of them, to prod them to fix their errors. To prod them to "do right by the students of the world" not their entrenched propaganda.
Length: 54 pages


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Amazon Best Sellers Rank: #253,425 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
#38 in 90-Minute Science & Math Short Reads
#318 in Calculus (Books)
#48 in Calculus (Kindle Store)





#4-5, 72nd published book

TEACHING TRUE MATHEMATICS: Volume 4 for age 19-20 Sophomore-year College, math textbook series, book 5 Kindle Edition
by Archimedes Plutonium (Author)

This is volume 3, series 4 of Teaching True Mathematics, designed for College Sophomore-year students, students of age 19-20. It is the continuation of volume 2 in the end-goal of learning how to do the mathematics of electricity and magnetism, because everything in physics is nothing but atoms and atoms are nothing but electricity and magnetism. To know math, you have to know physics. We learned the Calculus of 2nd dimension and applied it to the equations of physics for electricity and magnetism. But we did not learn the calculus of those equations for 3rd dimension. So, you can say that Sophomore year College math is devoted to 3D Calculus. This sophomore year college we fill in all the calculus, and we start over on all of Geometry, for geometry needs a modern day revision. And pardon me for this book is mostly reading, and the students doing less calculations. The classroom of this textbook has the teacher go through page by page to get the students comprehending and understanding of what is being taught. There are many hands on experiments also.

Cover Picture shows some toruses, some round some square, torus of rings, thin strips of rings or squares and shows them laid flat. That is Calculus of 3rd dimension that lays a ring in a torus to be flat in 2nd dimension.
Length: 103 pages

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#4-6, 75th published book

TEACHING TRUE MATHEMATICS: Volume 5 for age 20-21 Junior-year of College, math textbook series, book 6 Kindle Edition
by Archimedes Plutonium 2019


This is volume 4, series 5 of Teaching True Mathematics, designed for College Junior-year students, students of age 20-21. In first year college Calculus we learned calculus of the 2nd dimension and applied it to the equations of physics for electricity and magnetism. And in sophomore year we learned calculus of 3rd dimension to complete our study of the mathematics needed to do the physics of electricity and magnetism. Now, junior year college, we move onto something different, for we focus mostly on logic now and especially the logic of what is called the "mathematical proof". Much of what the student has learned about mathematics so far has been given to her or him as stated knowledge, accept it as true because I say so. But now we are going to do math proofs. Oh, yes, we did prove a few items here and there, such as why the Decimal Grid Number system is so special, such as the Pythagorean Theorem, such as the Fundamental Theorem of Calculus with its right-triangle hinged up or down. But many ideas we did not prove, we just stated them and expected all students to believe them true. And you are now juniors in college and we are going to start to prove many of those ideas and teach you "what is a math proof". Personally, I myself feel that the math proof is overrated, over hyped. But the math proof is important for one reason-- it makes you better scientists of knowing what is true and what is a shaky idea. A math proof is the same as "thinking straight and thinking clearly". And all scientists need to think straight and think clearly. But before we get to the Mathematics Proof, we have to do Probability and Statistics. What you learned in Grade School, then High School, then College, called Sigma Error, now becomes Probability and Statistics. It is important because all sciences including mathematics needs and uses Probability and Statistics. So, our job for junior-year of college mathematics is all cut out and ahead for us, no time to waste, let is get going.

Cover Picture: is a sample of the Array Proof, a proof the ellipse is not a conic but rather a cylinder cut wherein the oval is the slant cut of a cone, not the ellipse.
Length: 161 pages

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Publication Date: December 26, 2019

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#4-7, 89th published book

TEACHING TRUE MATHEMATICS: Volume 6 for age 21-22 Senior-year of College, math textbook series, book 7 Kindle Edition
by Archimedes Plutonium 2020

This is the last year of College for mathematics and we have to mostly summarize all of mathematics as best we can. And set a new pattern to prepare students going on to math graduate school. A new pattern of work habits, because graduate school is more of research and explore on your own. So in this final year, I am going to eliminate tests, and have it mostly done as homework assignments.

There is no better place to start any hard core science, even mathematics with the largest summary of all.

1) All is Atom and atoms are nothing but electricity and magnetism.

2) Electricity and Magnetism, EM, is based on a famous math equation of Voltage = i*B*L where i is current, B is magnetic field, and L is angular momentum. That is New Ohm's Law.

3) All the rest of physics and math equations and formulas of EM come from V=i*B*L

4) One of the most famous and great laws of electricity and magnetism that comes from V=i*B*L is likely to be the Faraday law. Remember its math in 2nd dimension? It is a division differential equation of voltage divided by the quantity of magnetic field times angular momentum. On the face of it, it is rather a simple equation, when you consider it involves the entire universe, both living and nonliving.

Cover Picture: Again and again, many times in math, the mind is not good enough alone to think straight and clear, and you need tools to hands-on see how it works. Here is a collection of tools for this senior year college classes. There is a pencil, clipboard, graph paper, compass, divider, protractor, slide-ruler. And for this year we spend a lot of time on the parallelepiped, showing my wood model, and showing my erector set model held together by wire loops in the corners. The plastic square is there only to hold up the erector set model.

Length: 109 pages

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#4-8, 90th published book

TEACHING TRUE MATHEMATICS: Volume 7 for age 22-26 Graduate school, math textbook series, book 8 Kindle Edition
by Archimedes Plutonium 2020

This is College Graduate School mathematics. Congratulations, you made it this far. To me, graduate school is mostly research, research mathematics and that means also physics. So it is going to be difficult to do math without physics. Of course, we focus on the mathematics of these research projects.

My textbook for Graduate school is just a template and the professors teaching the graduate students are free of course to follow their own projects, but in terms of being physics and math combined. What I list below is a template for possible projects.

So, in the below projects I list 31 of them and I expect students with their professor, to pick a project and to monitor the student as he/she progresses through the research. I have listed each project then cited some of my own research into these projects, below each project is an entry. Those entries are just a help or helper in getting started or acquainted with the project. The entry has a date time group and a newsgroup that I posted to such as sci.math or plutonium-atom-universe Google newsgroups. Again the entry is just a help or helper in getting started.

Now instead of picking one or two projects for your Graduate years of study, some may select all 31 projects where you write a short paper on each project. Some may be bored with just one or two projects and opt for all 31.

Cover Picture: A photo by my iphone of a page on Permutations of the Jacobs book Mathematics: A Human Endeavor, 1970. One of the best textbooks ever written in Old Math, not for its contents because there are many errors, but for its teaching style. It is extremely rare to find a math textbook written for the student to learn. Probably because math professors never learned how to teach in the first place; only learned how to unintentionally obfuscate. The page I photographed is important because it is the interface between geometry's perimeter or surface area versus geometry's area or volume, respectively. Or, an interface of pure numbers with that of geometry. But I have more to say on this below.
Length: 167 pages

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Publication Date: March 1, 2020
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[bassam karzeddin]

On Monday, October 19, 2020 at 6:05:26 PM UTC+3, Dan Christensen wrote:
> Recall: PROOF BY INDUCTION FOR 1 VARIABLE
>

> Suppose you want to prove that, for all m in N, we have P(m) being true.

>
> It will be sufficient to prove:
>

> 1. P(0)
>
> 2. For all k in N, we have: P(k) => P(k+1)
>
>
> *****************************************************************************

>
> PROOF BY INDUCTION FOR 2 VARIABLES
>
> Suppose you want to prove that, for all m, n in N, we have P(m, n) being true.
>
> It will be sufficient to prove:
>
> 1. P(0,0)
>
> 2. For all k in N, we have: P(0, k) => P(0, k+1)
>
> 3. For all j, k in N, we have: P(j, k) => P(j+1, k)
>
>

> Your comments?

>
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

This is the type of fart business (mathematics, logic, philosophy and physics as well) introduced solely by Trolls and Imbeciles of truer anti-free human knowledge

They simply can't prove anything as you can see from their thousands of same threads because they refuse to understand the most simple facts about their entire false foundations

How can they do something meaningful if they are still very clueless and can't and don't want to understand the simplest oldest impossible construction problems raised by the ancient Greeks (by any means or by any tools, I added)

Those are the true anti-free human knowledge who are living entirely in the fool's Paradises and refuse to come out of it

But in all cases, they are not EVEN noticeable nor comparable to even a skilled footballer boy of today's world FOR SURE

bkk

[zelos...@gmail.com]

måndag 19 oktober 2020 kl. 17:25:32 UTC+2 skrev Dan Christensen:
> On Monday, October 19, 2020 at 11:22:58 AM UTC-4, FredJeffries wrote:

> > On Monday, October 19, 2020 at 8:05:26 AM UTC-7, Dan Christensen wrote:
> > > Recall: PROOF BY INDUCTION FOR 1 VARIABLE
> > >
> > > Suppose you want to prove that, for all m in N, we have P(m) being true.
> > >
> > > It will be sufficient to prove:
> > >
> > > 1. P(0)
> > >
> > > 2. For all k in N, we have: P(k) => P(k+1)
> > >
> > >
> > > *****************************************************************************
> > >
> > > PROOF BY INDUCTION FOR 2 VARIABLES
> > >
> > > Suppose you want to prove that, for all m, n in N, we have P(m, n) being true.
> > >
> > > It will be sufficient to prove:
> > >
> > > 1. P(0,0)
> > >
> > > 2. For all k in N, we have: P(0, k) => P(0, k+1)
> > >
> > > 3. For all j, k in N, we have: P(j, k) => P(j+1, k)
> > >
> > >
> > > Your comments?
> >

> > Congratulations. You have just proved that for all natural numbers j, k, the product j*k = 0

> Think again.
Fred is correct unfortunately

[Mostowski Collapse]

Thats also not induction along omega^3. Seems
you are ignorant of well orderings?

Well orderings have to be linear orders. Another
name for linear order is total order.

https://en.wikipedia.org/wiki/Total_order

Means the ordering has to have this law:

a =< b \/ b =< a

i.e. every pair of elements needs to be comparable.
For your 2 arguments ordering you already don't

have linearity. This means it makes hard if not
impossible to carry out certain inductive proofs.

Same problem with your 3 arguments ordering.

[Mostowski Collapse]

The way you move **forward** on the grid,
through your Dan-O-Matic 2 argument

induction, does for example not allow
that one of these pairs can reach the

other pair:

<3,4> ?? <4,5>

If you go forward from <3,4> you
get <4,4>, <5,4>, etc.. and <4,5>
is not among them.

If you go forward from <4,5> you
get <5,5>, <6,4>, etc.. and <3,4>
is not among them.

So your induction is not based on
a linear order. Means you cannot use
it to prove for example:

/* P(u,v) :<=> u+(v+1) = (u+1)+v */

forall u,v (u+(v+1) = (u+1)+v)

Please prove the above in DC Troll Proof.

[Graham Cooper]

Prove

j*k=0 -> (j+1)*k = 0

[Dan Christensen]

On Wednesday, October 21, 2020 at 4:26:00 AM UTC-4, Mostowski Collapse wrote:
> Thats also not induction along omega^3.

Not sure what you are getting at here, Jan Burse, but this proof can be applied on any set n on which induction holds. Note that it does not make use of Peano's 3rd and 4th axioms. So n may even be finite.

For induction to hold on a set n, we required only n0 in n and function S: n --> n such that

n = {n0, S(n0), S(S(n0)), S(S(S(n0))), ... }

In other words, every element of n is accessible from n0 by a process of repeated succession under S.

For a formal proof, see https://www.dcproof.com/InductionIffAccessible.htm (228 lines) at my math blog.

[Archimedes Plutonium]

Jan Burse & Dan Christensen, fools & idiots of logic with their 10 OR 2 = 12 and their AND as subtraction, they do not belong in sci.math but in a alt kook spittle spattle

5th published book of AP

Suspend all College Classes in Logic, until they Fix their Errors // Logic series, book 1 Kindle Edition
by Archimedes Plutonium (Author)

First comes Logic-- think straight and clear which many logic and math professors are deaf dumb and blind to, and simply refuse to recognize and fix their errors.

The single biggest error of Old Logic of Boole and Jevons was their "AND" and "OR" connectors. They got them mixed up and turned around. For their logic ends up being that of 3 OR 2 = 5 with 3 AND 2 = 1, when even the local village idiot knows that 3 AND 2 = 5 (addition) with 3 OR 2 = either 3 or 2 (subtraction). And secondly, their error of the If->Then conditional. I need to make it clear enough to the reader why the true Truth Table of IF --> Then requires a U for unknown or uncertain with a probability outcome for F --> T = U and F --> F = U. Some smart readers would know that the reason for the U is because without the U, Logic has no means of division by 0 which is undefined in mathematics. You cannot have a Logic that is less than mathematics. A logic that is impoverished and cannot do a "undefined for division by 0 in mathematics". The true logic must be able to have the fact that division by 0 is undefined. True logic is larger than all of mathematics, and must be able to fetch any piece of mathematics from out of Logic itself. So another word for U is undefined. And this is the crux of why Reductio ad Absurdum cannot be a proof method of mathematics, for a starting falsehood in a mathematics proof can only lead to a probability end conclusion.

My corrections of Old Logic have a history that dates before 1993, sometime around 1991, I realized the Euclid proof of infinitude of primes was illogical, sadly sadly wrong, in that the newly formed number by "multiply the lot and add 1" was necessarily a new prime in the indirect proof method. So that my history of fixing Old Logic starts in 1991, but comes to a synthesis of correcting all four of the connectors of Equal/not, And, Or, If->Then, by 2015.

Cover picture: some may complain my covers are less in quality, but I have a good reason for those covers-- I would like covers of math or logic to show the teacher's own handwriting as if he were back in the classroom writing on the blackboard or an overhead projector.
Length: 65 pages

File Size: 764 KB
Print Length: 65 pages

Publication Date: March 12, 2019

Sold by: Amazon Digital Services LLC
Language: English

ASIN: B07PMB69F5

[Dan Christensen]

Another troll heard from!

On Wednesday, October 21, 2020 at 3:37:07 AM UTC-4, bassam karzeddin (aka BKK) wrote:

> This is the type of fart business (mathematics, logic, philosophy and physics as well) introduced solely by Trolls and Imbeciles of truer anti-free human knowledge
>

From Psycho Troll BKK who also wrote here:

“Those many challenges of mine (in my posts) weren't actually designed for human beings, but for the future artificial beings that would certainly replace them not far away from now, for sure.”
-- BKK, Dec. 6, 2017

"The Devils deeds that are strictly and basically sourced from mathematicians like humans, FOR SURE!"
-- BKK, June 11, 2020

“You know certainly that I'm the man, and more specially the KING who is going to upside down most of your current false mathematics for all future generations.”
-- BKK, Nov. 22, 2018

“Despite thousands of years of continuous juggling and false definitions of what is truly the real number, they [us carbon-based lifeforms?] truly don't want to understand it as was discovered strictly by the *KING* [BKK Himself!]”
-- BKK, Nov. 28, 2019

“I don't believe even in one being a number”
-- BKK, Dec. 31, 2019

Math failure, BKK, doesn't believe in negative numbers, zero, one or numbers like pi and root 2. He doesn't even believe in 40 degree angles or circles. Simple speed-distance-time problems seem to be impossible for him. Really!

Needless to say his own goofy little system is getting nowhere and never will. As such he is insanely jealous of wildly successful mainstream mathematics. He seems to believe these super-intelligent artificial beings of his will somehow be enlisting his aid to "reform" mathematics worldwide when they take over the planet in the near future. He is truly delusional.

[Dan Christensen]

On Wednesday, October 21, 2020 at 12:43:54 AM UTC-4, Archimedes Plutonium wrote:

>
>
> Learn the 50 fakes and mistakes of Archimedes Plutonium's FAKE Math...

[Peter]

Dan Christensen wrote:
> On Monday, October 19, 2020 at 11:05:26 AM UTC-4, Dan Christensen wrote:
>
>> PROOF BY INDUCTION FOR 2 VARIABLES
>>
>> Suppose you want to prove that, for all m, n in N, we have P(m, n) being true.
>>
>> It will be sufficient to prove:
>>
>> 1. P(0,0)
>>
>> 2. For all k in N, we have: P(0, k) => P(0, k+1)
>>
>> 3. For all j, k in N, we have: P(j, k) => P(j+1, k)
>>
>>
>
> And now...
>
> Proof by Induction with *** THREE *** Variables
>
>
> Suppose you want to prove that P(x, y, z) is true for all x, y, z in N. It will suffice to prove each of the following?
>
> 1. P(0, 0, 0)
>
> 2. For all k in N:[P(0, 0, k) => P(0, 0, k+1)]
>
> 3. For all j, k in N:[P(0, j, k) => P(0, j+1, k)]
>
> 4. For all i, j, k in N:[P(i,j,k) => P(i+1,j ,k)

You should be able to prove your two- and three-variable inductions from
one-variable induction by imposing a suitable order on ordered pairs of
natural numbers.

Consider p:NxN -> N defined by

p(x,y) = (x+y)(x+y+1)/2

which is invertible (see https://en.wikipedia.org/wiki/Pairing_function,
for example). That enables you to order the elements of NxN.

By defining the pair (x,y,z) to be ((x,y),z) an order can also be
imposed on NxNxN.

> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
>

--
When, once, reference was made to a statesman almost universally
recognized as one of the villains of this century, in order to
induce him to a negative judgment, he replied: "My situation is
so different from his, that it is not for me to pass judgment".
Ernst Specker on Paul Bernays

[Dan Christensen]

In theory, I guess, but having written several rather long inductive proofs with multiple (often 3) variables the hard way using regular induction, it seems to me that your suggestion would probably not help matters. Thanks anyway.

[zelos...@gmail.com]

onsdag 21 oktober 2020 kl. 21:25:21 UTC+2 skrev Archimedes Plutonium:
> Dan Christensen is a shithead of math and does not deserve to post in sci.math for he only entrenches fake math fake logic.
>
> Time is up on the two insane logic of Dan Christensen & Jan Burse, and to exit sci.math with their b.s. of 10 OR 2 = 12 and their AND as subtraction.
>
> Archimedes Plutonium
>
> to
> Learn true calculus, not the idiotic mindless Old Math calculus with their mindless LIMIT barbwired contraption. Only a insane idiot would think that a limit argument serves as a proof of mathematics. For the LIMIT argument was invented by Cauchy to dumbnify his smartest students who saw through Cauchy's failed calculus.
>
> Cauchy and Old Math want you to believe you can have a rectangle of 0 width and still have interior area. And that is what Old Math professors teach-- integral is the sum of rectangles of 0 width.
>
> AP

You are speaking about yourself mate

[Archimedes Plutonium]

[zelos...@gmail.com]

You are still wrong and have failed basic mathematics

[Mostowski Collapse]

Dan-O-Matic, bigfoot from Canada, can
you prove this here:

forall u,v (u+(v+1) = (u+1)+v)

with your Botched Proof by Induction
Schema with 2 Variables? In DC Troll Proof?

[Peter]

Does DC Proof prove induction on well-founded relations?

[Dan Christensen]

Not sure if this answers your question, but I have usually been using the following axiom for the Axiom of Induction (Peano's 5th Axiom):

ALL(a):[Set(a) & ALL(b):[b in a => b in n]

=> [0 in a & ALL(b):[b in a => s(b) in a] => ALL(b):[b in n => b in a]]]

As I have mentioned here, induction can also be shown to hold not just on N, but other sets as well, even on finite sets.

[Mostowski Collapse]

Nope, DC Troll Proof thinks it doesn't
need the regularity axiom,

since Halmos omitted it as well.


LoL

[Dan Christensen]

On Thursday, October 22, 2020 at 2:09:51 PM UTC-4, Mostowski Collapse wrote:
> Nope, DC Proof thinks it doesn't
> need the regularity axiom,
>

If for some reason, you actually wanted a regularity axiom, you could introduce one at the beginning of your proof. Something like:

ALL(a):[Set(a) & EXIST(b):b in a => EXIST(c):[Set(c) & c in a & ALL(d):[d in a => ~d in c]]]

It's so, ummm... obvious and self-evident!! :^P

[Archimedes Plutonium]

Why kibo Parry Moroney calls Dan Christensen & sidekick Jan Burse Village idiots and math failures:

>"Village Idiot"
>flunked the Math Test of a lifetime-generation test
On Thursday, October 22, 2020 at 9:01:34 AM UTC-5, Michael Moroney wrote:
> Math Failure
> Wow, a new Megatard Moment in just two days! Excellent work, Failure

AP writes: Is it because Dan and Jan cannot tell the difference between ellipse and oval? Or is it because Dan & Jan think 10 OR 2 = 12 with AND as subtraction?
I think it is because Dan is such a stupid idiot in all of science as witnessed by his comment of distinct verus nondistinct:

On Wednesday, January 25, 2017 at 10:08:09 AM UTC-6, Peter Percival wrote:
> Dan Christensen wrote:

> > On Wednesday, January 25, 2017 at 9:47:32 AM UTC-5, Archimedes Plutonium wrote:
> >> On Wednesday, January 25, 2017 at 8:27:19 AM UTC-6, Dan Christensen wrote:
> >>> On Wednesday, January 25, 2017 at 9:16:52 AM UTC-5, Archimedes Plutonium wrote:
> >>>> PAGE58, 8-3, True Geometry / correcting axioms, 1by1 tool, angles of logarithmic spiral, conic sections unified regular polyhedra, Leaf-Triangle, Unit Basis Vector
> >>>>
> >>>> The axioms that are in need of fixing is the axiom that between any two points lies a third new point.
> >>>
> >>> The should be "between and any two DISTINCT points."
> >>>
> >>
> >> What a monsterous fool you are
> >>
> >
> > OMG. You are serious. Stupid and proud of it.
>
> And yet Mr Plutonium is right.  Two points are distinct (else they would
> be one) and it is not necessary to say so.
>

[Mostowski Collapse]

Ha Ha, from disinformation to pure hate.
Micro penis shows his true face.

Why dont you ask the Bill & Melinda
foundation for some brains, they might

upload it through some nose swab.

LoL

On Tuesday, October 20, 2020 at 5:01:20 PM UTC+2, Brody Roccanello wrote:
> Mostowski Collapse wrote:
>
> > Possibly the Dan-O-Matic ordering is not linear. It is neither:
> > <3,4> <_1 <4,5> Nor: <4,5> <_1 <3,4> Also not transitively?
>
> the stupidest lines I ever see. But your *covid-19* is no longer a
> pandemic, but a *_syndemic_*, which is about obesity, sedentary and
> climate change. No joke, when you move around. What can you say in your
> defence.
>
> https://www.marca.com/en/lifestyle/
> 2020/10/15/5f87777c268e3ec9778b45ac.html
>
> https://www.thelancet.com/journals/lancet/article/
> PIIS0140-6736(20)32000-6/fulltext

[Peter]

Peter wrote:


> Does DC Proof prove induction on well-founded relations?

There is this beautiful theorem (due, I think to Montague):

If, for all well-founded sets x, if P(y) for all y in x, then P(x),
then P(z) for all well-founded sets z.

[Mostowski Collapse]

It helps proving commutativity first.

Lemma.
m+s(n)=s(m)+n for every m,n in N

Theorem.
m+n=n+m for every m,n in N.

https://www.dpmms.cam.ac.uk/~wtg10/addcomm.html

LoL

Is the above proof correct?

[Mostowski Collapse]

Shit ~wgt10 is William Timothy Gowers.
Looks like his student webpage.

LoL

[Archimedes Plutonium]

Is this the Univ Western Ontario Dan Christensen, who keeps stalking AP for 7 years now? So stupid in Logic with his 10 OR 2 = 12? And his inability to even square away "distinct from nondistinct"?

On Wednesday, January 25, 2017 at 10:08:09 AM UTC-6, Peter Percival wrote:
> Dan Christensen wrote:
> > On Wednesday, January 25, 2017 at 9:47:32 AM UTC-5, Archimedes Plutonium wrote:
> >> On Wednesday, January 25, 2017 at 8:27:19 AM UTC-6, Dan Christensen wrote:
> >>> On Wednesday, January 25, 2017 at 9:16:52 AM UTC-5, Archimedes Plutonium wrote:
> >>>> PAGE58, 8-3, True Geometry / correcting axioms, 1by1 tool, angles of logarithmic spiral, conic sections unified regular polyhedra, Leaf-Triangle, Unit Basis Vector
> >>>>
> >>>> The axioms that are in need of fixing is the axiom that between any two points lies a third new point.
> >>>
> >>> The should be "between and any two DISTINCT points."
> >>>
> >>
> >> What a monsterous fool you are
> >>
> >
> > OMG. You are serious. Stupid and proud of it.
>
> And yet Mr Plutonium is right.  Two points are distinct (else they would
> be one) and it is not necessary to say so.
>

5th published book

[Dan Christensen]

On Monday, October 26, 2020 at 2:52:23 PM UTC-4, Archimedes Plutonium wrote:
> Is this the ...

[Graham Cooper]

On Wednesday, October 21, 2020 at 5:52:25 PM UTC+10, zelos...@gmail.com wrote:
> måndag 19 oktober 2020 kl. 17:25:32 UTC+2 skrev Dan Christensen:
> > On Monday, October 19, 2020 at 11:22:58 AM UTC-4, FredJeffries wrote:
> > > On Monday, October 19, 2020 at 8:05:26 AM UTC-7, Dan Christensen wrote:
> > > > Recall: PROOF BY INDUCTION FOR 1 VARIABLE
> > > >

> > > > Suppose you want to prove that, for all m in N, we have P(m) being true.

> > > >
> > > > It will be sufficient to prove:
> > > >

> > > > 1. P(0)
> > > >
> > > > 2. For all k in N, we have: P(k) => P(k+1)
> > > >
> > > >
> > > > *****************************************************************************

> > > >
> > > > PROOF BY INDUCTION FOR 2 VARIABLES
> > > >
> > > > Suppose you want to prove that, for all m, n in N, we have P(m, n) being true.
> > > >
> > > > It will be sufficient to prove:
> > > >
> > > > 1. P(0,0)
> > > >
> > > > 2. For all k in N, we have: P(0, k) => P(0, k+1)
> > > >
> > > > 3. For all j, k in N, we have: P(j, k) => P(j+1, k)
> > > >
> > > >

> > > > Your comments?
> > >
> > > Congratulations. You have just proved that for all natural numbers j, k, the product j*k = 0
> > Think again.
> Fred is correct unfortunately

NO YOUR JUST A STUPID FUKING IDIOT!

[Archimedes Plutonium]

2- Is this the Univ Western Ontario Dan Christensen, who keeps stalking AP for 7 years now? So stupid in Logic with his 10 OR 2 = 12? And his inability to even square away "distinct from nondistinct"?

#6-2, 27th published book

Correcting Reductio Ad Absurdum// Logic series, book 2 Kindle Edition
by Archimedes Plutonium (Author)


These are the TRUE Truth Tables of the 4 connectors of Logic

Equal+Not                    
T = T  =  T                      
T = ~F = T                      
F = ~T = T
F = F   = T   

If--> then                  
T --> T  = T
T --> F  = F
F --> T  = U  (unknown or uncertain)           
F --> F  = U  (unknown or uncertain)

And

T  &  T = T                       
T  &  F = T                      
F  &  T = T                      
F  &  F = F                      

Or
T  or  T  = F
T  or  F  = T
F  or  T  = T
F  or  F  = F

Those can be analyzed as being Equal+Not is multiplication, If-->then is division, And is addition and Or is subtraction in mathematics. Now I need to emphasis this error of Old Logic, the If->Then conditional. I need to make it clear enough to the reader why the true Truth Table of IF --> Then requires a U for unknown or uncertain with a probability outcome for F --> T = U and F --> F = U. Some smart readers would know that the reason for the U is because without the U, Logic has no means of division by 0 which is undefined in mathematics. You cannot have a Logic that is less than mathematics. A logic that is impoverished and cannot do a "undefined for division by 0 in mathematics". The true logic must be able to have the fact that division by 0 is undefined. True logic is larger than all of mathematics, and must be able to fetch any piece of mathematics from out of Logic itself. So another word for U is undefined. And this is the crux of why Reductio ad Absurdum cannot be a proof method of mathematics, for a starting falsehood in a mathematics proof can only lead to a probability end conclusion.

Now in Old Logic they had for Reductio Ad Absurdum as displayed by this schematic:

|    | ~p
|    |---
|    | .
|    | .
|    | q
|    | .
|    | .
|    | ~q
| p

Which is fine except for the error of not indicating the end conclusion of "p" is only a probability of being true, not guaranteed as true. And this is the huge huge error that mathematicians have fallen victim of. For the Reductio Ad Absurdum is not a proof method for mathematics, it is probability of being true or false. Math works on guaranteed truth, not probability. This textbook is written to fix that error.

Cover Picture: I like my covers to be like as if a blackboard in school to connect with students. This is a picture of the above Reductio Ad Absurdum, as a student or teacher would write in their notes or blackboard.
Length: 82 pages

File Size: 1175 KB
Print Length: 82 pages
Publication Date: March 23, 2019

Sold by: Amazon Digital Services LLC
Language: English

ASIN: B07Q18GQ7S

Text-to-Speech: Enabled 
X-Ray: 
Not Enabled  

Word Wise: Not Enabled
Lending: Enabled

Enhanced Typesetting: Enabled 




#6-3, 143rd published book

DeMorgan's Laws are fantasies, not laws// Logic series, book 3 Kindle Edition
By Archimedes Plutonium

The Logic community never had the correct truth table of the primitive 4 connectors of Logic, (1) Equal compounded with NOT, (2) AND, (3) OR, (4) IF->THEN. In 1800s, the founders of Logic messed up in terrible error all 4 of the primitive logic connectors. And since the 1900s, AP was wanted an explanation why Old Logic got all 4 connectors in total error? What was the reason for the mess up? And in the past few years, I finally pinned the reason to starting Logic with DeMorgan's fake laws, from which Boole, a close friend of DeMorgan, was going to keep his friendship and accept the DeMorgan Laws. That meant that DeMorgan, Boole, Jevons accepted OR as being that of Either..Or..Or..Both, what is called the inclusive OR. But the inclusive OR is a contradiction in terms, for there never can exist a combo of OR with AND simultaneously. This book goes into detail why the DeMorgan laws are fake and fantasy.

Product details
• File Size : 620 KB
• Word Wise : Enabled
• Print Length : 38 pages
• ASIN : B08M4BY4XM
• Publication Date : October 27, 2020
• Language: : English
• Enhanced Typesetting : Enabled
• X-Ray : Not Enabled
• Text-to-Speech : Enabled
• Screen Reader : Supported
• Lending : Enabled
•
•





#6-4, 100th published book

Pragmatism, the only Philosophy I loved // Logic series, book 4 Kindle Edition
By Archimedes Plutonium

I need to give credit to the philosophy of Pragmatism, the only philosophy that I know of that is based on science. Credit for my discovery of the Plutonium Atom Totality in 1990, came in part, partially due to a passage of the Pragmatist Charles Sanders Peirce in Peirce's Cosmology:

 Peirce's The Architecture of Theories...
         ...would be a Cosmogonic Philosophy. It would suppose that in the beginning - infinitely remote - there was a chaos of unpersonalized feeling, which being without connection or regularity would properly be without existence. This feeling, sporting here and there in pure arbitrariness, would have
started the germ of a generalizing tendency. Its other sportings would be evanescent, but this would have a growing virtue. Thus, the tendency to habit would be started; and from this, with the other principles of evolution, all the regularities of the universe would be evolved. At any time, however, an element of pure chance survives and will remain until the world becomes an absolutely perfect, rational, and symmetrical system, in which mind is at last crystallized in the infinitely distant future.
--- end quoting Peirce's Cosmology ---

But also I must give credit to Pragmatism for making it a philosophy one can actually live their lives by, for living a life of pragmatic solutions to everyday problems that occur in my life. A case in point example is now in March 2020, being the pragmatist that I am, and enduring the 2020 corona virus pandemic. No other philosophy that I know of is so keenly in tune with a person, the surrounding environment and how to live.

Cover Picture: is my photograph of the two books on Pragmatism that I almost looked upon as my-bibles-of-living. The small paperback was bought by me at Univ Cincinnati when I was a 1st year student 1969 not for any course, but my own self study reading, for I was curious of what philosophies existed. The second one I purchased at Melbourne Univ, 1974, when I was a math teacher in Australia. So good was the 1969 book that I bought the 1974 book, not knowing until 1990 how crucial and critical was this book with its Peirce's Cosmology.
Length: 111 pages

Product details
File Size: 800 KB
Print Length: 111 pages
Publication Date: March 14, 2020
Sold by: Amazon.com Services LLC
Language: English
ASIN: B085X863QW

Text-to-Speech: Enabled 
X-Ray: 
Not Enabled  


Word Wise: Enabled

[Mostowski Collapse]

Archimedes Plutonium should be thrown in jail
for his willful criminal behavior. The criminal
Archimedes Plutonium all the times posts people
name lists together with hate speach about these people.

It is highly likely Archimedes Plutonium is
psycho. Archimedes Plutonium belongs in prison not
on usenet for his mind is complete hate hate hate.
Put the creep in jail and throw away the keys.