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How to define '=' in expression "a=b"?

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  How to define '=' in expression "a=b"? wij <wyniijj@gmail.com> - 2021-07-18 18:21 -0700
    Re: How to define '=' in expression "a=b"? Charlie-Boo <shymathguy@gmail.com> - 2021-07-19 08:00 -0700
    Re: How to define '=' in expression "a=b"? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-07-19 16:06 +0100
      Re: How to define '=' in expression "a=b"? wij <wyniijj@gmail.com> - 2021-07-19 10:24 -0700
        Re: How to define '=' in expression "a=b"? Peter <peterxpercival@hotmail.com> - 2021-07-19 18:44 +0100
          Re: How to define '=' in expression "a=b"? wij <wyniijj@gmail.com> - 2021-07-19 18:28 -0700
            Re: How to define '=' in expression "a=b"? Peter <peterxpercival@hotmail.com> - 2021-07-22 13:22 +0100
              Re: How to define '=' in expression "a=b"? wij <wyniijj@gmail.com> - 2021-07-22 07:25 -0700
                Re: How to define '=' in expression "a=b"? Peter <peterxpercival@hotmail.com> - 2021-07-22 17:20 +0100
        Re: How to define '=' in expression "a=b"? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-07-19 21:45 +0100
          Re: How to define '=' in expression "a=b"? wij <wyniijj@gmail.com> - 2021-07-19 18:30 -0700
            Re: How to define '=' in expression "a=b"? Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-07-20 11:18 +0100
        Re: How to define '=' in expression "a=b"? Charlie-Boo <shymathguy@gmail.com> - 2021-07-22 08:26 -0700
          Re: How to define '=' in expression "a=b"? wij <wyniijj@gmail.com> - 2021-07-22 09:41 -0700
      Re: How to define '=' in expression "a=b"? Charlie-Boo <shymathguy@gmail.com> - 2021-07-22 08:19 -0700
      Re: How to define '=' in expression "a=b"? Charlie-Boo <shymathguy@gmail.com> - 2021-07-22 08:51 -0700
        Re: How to define '=' in expression "a=b"? Charlie-Boo <shymathguy@gmail.com> - 2021-07-22 08:52 -0700
    Re: How to define '=' in expression "a=b"? Peter <peterxpercival@hotmail.com> - 2021-07-19 18:28 +0100
      Re: How to define '=' in expression "a=b"? wij <wyniijj@gmail.com> - 2021-07-19 18:27 -0700
        Re: How to define '=' in expression "a=b"? Charlie-Boo <shymathguy@gmail.com> - 2021-07-22 08:30 -0700
      Re: How to define '=' in expression "a=b"? Charlie-Boo <shymathguy@gmail.com> - 2021-07-22 08:23 -0700
    Re: How to define '=' in expression "a=b"? Jeff Barnett <jbb@notatt.com> - 2021-07-19 14:41 -0600
      Re: How to define '=' in expression "a=b"? wij <wyniijj@gmail.com> - 2021-07-19 19:52 -0700
        Re: How to define '=' in expression "a=b"? Jeff Barnett <jbb@notatt.com> - 2021-07-19 22:04 -0600
          Re: How to define '=' in expression "a=b"? wij <wyniijj@gmail.com> - 2021-07-21 20:14 -0700
        Re: How to define '=' in expression "a=b"? Peter <peterxpercival@hotmail.com> - 2021-07-22 13:24 +0100
          Re: How to define '=' in expression "a=b"? wij <wyniijj@gmail.com> - 2021-07-22 07:26 -0700
            Re: How to define '=' in expression "a=b"? Peter <peterxpercival@hotmail.com> - 2021-07-22 17:16 +0100
    Re: How to define '=' in expression "a=b"? olcott <NoOne@NoWhere.com> - 2021-07-22 10:17 -0500
      Re: How to define '=' in expression "a=b"? wij <wyniijj@gmail.com> - 2021-07-22 09:31 -0700
        Re: How to define '=' in expression "a=b"? olcott <NoOne@NoWhere.com> - 2021-07-22 11:44 -0500
          Re: How to define '=' in expression "a=b"? wij <wyniijj@gmail.com> - 2021-07-22 09:52 -0700
            Re: How to define '=' in expression "a=b"? olcott <NoOne@NoWhere.com> - 2021-07-22 12:01 -0500
              Re: How to define '=' in expression "a=b"? Peter <peterxpercival@hotmail.com> - 2021-07-22 19:13 +0100
                Re: How to define '=' in expression "a=b"? Richard Damon <Richard@Damon-Family.org> - 2021-07-22 11:24 -0700
                Re: How to define '=' in expression "a=b"? [ Gödel as Tarski ] olcott <NoOne@NoWhere.com> - 2021-07-24 13:18 -0500
              Re: How to define '=' in expression "a=b"? Peter <peterxpercival@hotmail.com> - 2021-07-24 18:57 +0100
                Re: How to define '=' in expression "a=b"? Jeff Barnett <jbb@notatt.com> - 2021-07-24 12:12 -0600
                  Re: How to define '=' in expression "a=b"? olcott <NoOne@NoWhere.com> - 2021-07-24 13:33 -0500
                    Re: How to define '=' in expression "a=b"? Jeff Barnett <jbb@notatt.com> - 2021-07-24 15:56 -0600
                      Re: How to define '=' in expression "a=b"? olcott <NoOne@NoWhere.com> - 2021-07-24 17:05 -0500
                        Re: How to define '=' in expression "a=b"? Jeff Barnett <jbb@notatt.com> - 2021-07-24 16:13 -0600
                        Re: How to define '=' in expression "a=b"? Richard Damon <Richard@Damon-Family.org> - 2021-07-24 19:52 -0700
                Re: How to define '=' in expression "a=b"? ? [ Gödel as Tarski ] olcott <NoOne@NoWhere.com> - 2021-07-24 13:27 -0500
                  Re: How to define '=' in expression "a=b"? ? [ Gödel as Tarski ] Peter <peterxpercival@hotmail.com> - 2021-07-25 12:59 +0100
                    Re: How to define '=' in expression "a=b"? ? [ Gödel as Tarski ] olcott <NoOne@NoWhere.com> - 2021-07-25 10:13 -0500
                      Re: How to define '=' in expression "a=b"? ? [ Gödel as Tarski ] André G. Isaak <agisaak@gm.invalid> - 2021-07-25 09:55 -0600
                        Re: How to define '=' in expression "a=b"? ? [ Gödel as Tarski ] olcott <NoOne@NoWhere.com> - 2021-07-25 11:23 -0500
                          Re: How to define '=' in expression "a=b"? ? [ Gödel as Tarski ] André G. Isaak <agisaak@gm.invalid> - 2021-07-25 10:51 -0600
                            Re: How to define '=' in expression "a=b"? ? [ Gödel as Tarski ] olcott <NoOne@NoWhere.com> - 2021-07-25 12:02 -0500
                              Re: How to define '=' in expression "a=b"? ? [ Gödel as Tarski ] André G. Isaak <agisaak@gm.invalid> - 2021-07-25 11:40 -0600
                                Re: How to define '=' in expression "a=b"? ? [ Gödel as Tarski ] olcott <NoOne@NoWhere.com> - 2021-07-25 13:05 -0500
                                  Re: How to define '=' in expression "a=b"? ? [ Gödel as Tarski ] André G. Isaak <agisaak@gm.invalid> - 2021-07-25 13:07 -0600
          Re: How to define '=' in expression "a=b"? Richard Damon <Richard@Damon-Family.org> - 2021-07-22 10:28 -0700

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#36615 — How to define '=' in expression "a=b"?

Definition of equal sign '=' (mathematics ):
1. Basic definition: a=b iff a and b are exchangeable
2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable

Any suggestion improving the definition of '=' in mathematical expression?
(The axiom of extension in set theory is not appropriate, because it is set only)

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#36643

On Sunday, July 18, 2021 at 9:21:59 PM UTC-4, wij wrote:
> Definition of equal sign '=' (mathematics ): 
> 1. Basic definition: a=b iff a and b are exchangeable 
> 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable 
> 
> Any suggestion improving the definition of '=' in mathematical expression? 
> (The axiom of extension in set theory is not appropriate, because it is set only)

= is a license to substitute.

C-B

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#36645

wij <wyniijj@gmail.com> writes:

> Definition of equal sign '=' (mathematics ):
> 1. Basic definition: a=b iff a and b are exchangeable
> 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable
>
> Any suggestion improving the definition of '=' in mathematical
> expression?

It's a big topic, but any book on formal logic will explain the
definition used in first order theories.  You'll also find an
explanation of bound and free variables which is key to correctly
defining substitution in formulas.

-- 
Ben.

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#36652

On Monday, 19 July 2021 at 23:06:18 UTC+8, Ben Bacarisse wrote:
> wij <wyn...@gmail.com> writes: 
> 
> > Definition of equal sign '=' (mathematics ): 
> > 1. Basic definition: a=b iff a and b are exchangeable 
> > 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable 
> > 
> > Any suggestion improving the definition of '=' in mathematical 
> > expression?
> It's a big topic, but any book on formal logic will explain the 
> definition used in first order theories. You'll also find an 
> explanation of bound and free variables which is key to correctly 
> defining substitution in formulas. 
> 
> -- 
> Ben.

I quickly scanned all possible books related to your "formal logic",...none.
The '=' is basically used unexplained.

Usage example: (A little different than usual meaning. Just for example, I am
talking about how to define '='(equal) in average meaning commonly used)

Definition: G=x^2 or G=a&b
Modulo: a=b (mod n) 

Similar ones are:
x ⊢ y means y is derivable from x.
A ↔ B, A ⇔ B means A is true iff B is true 

-------------------------
To be more specific about the question.
If you are to define '=', how would you address it? or others.

Example:
1. lim(n->∞) 1/n=0  
2. (a+b)^2=a^2+2ab+b^2

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#36654

wij wrote:

> I quickly scanned all possible books related to your "formal logic",...none.
> The '=' is basically used unexplained.

I doubt it!


-- 
The world will little note, nor long remember what we say here
Abraham Lincoln at Gettysburg

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#36678

On Tuesday, 20 July 2021 at 01:44:22 UTC+8, Peter wrote:
> wij wrote: 
> 
> > I quickly scanned all possible books related to your "formal logic",...none. 
> > The '=' is basically used unexplained.
> I doubt it!
> -- 
> The world will little note, nor long remember what we say here 
> Abraham Lincoln at Gettysburg

Show what's in your book, it should be less than 2 lines.

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#36842

wij wrote:
> On Tuesday, 20 July 2021 at 01:44:22 UTC+8, Peter wrote:
>> wij wrote:
>>
>>> I quickly scanned all possible books related to your "formal logic",...none.
>>> The '=' is basically used unexplained.
>> I doubt it!
>> -- 
>> The world will little note, nor long remember what we say here
>> Abraham Lincoln at Gettysburg
> 
> Show what's in your book, it should be less than 2 lines.
> 
I have numerous books on logic.  It is more than two lines in the few I 
looked at before answering.

-- 
The world will little note, nor long remember what we say here
Abraham Lincoln at Gettysburg

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#36847

On Thursday, 22 July 2021 at 20:22:16 UTC+8, Peter wrote:
> wij wrote: 
> > On Tuesday, 20 July 2021 at 01:44:22 UTC+8, Peter wrote: 
> >> wij wrote: 
> >> 
> >>> I quickly scanned all possible books related to your "formal logic",...none. 
> >>> The '=' is basically used unexplained. 
> >> I doubt it! 
> >> -- 
> >> The world will little note, nor long remember what we say here 
> >> Abraham Lincoln at Gettysburg 
> > 
> > Show what's in your book, it should be less than 2 lines. 
> >
> I have numerous books on logic. It is more than two lines in the few I 
> looked at before answering.
> -- 
> The world will little note, nor long remember what we say here 
> Abraham Lincoln at Gettysburg

Yes, I found several plausible ones on internet.
But, which one can be used to define irrational numbers better?

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#36861

wij wrote:
> On Thursday, 22 July 2021 at 20:22:16 UTC+8, Peter wrote:
>> wij wrote:
>>> On Tuesday, 20 July 2021 at 01:44:22 UTC+8, Peter wrote:
>>>> wij wrote:
>>>>
>>>>> I quickly scanned all possible books related to your "formal logic",...none.
>>>>> The '=' is basically used unexplained.
>>>> I doubt it!
>>>> -- 
>>>> The world will little note, nor long remember what we say here
>>>> Abraham Lincoln at Gettysburg
>>>
>>> Show what's in your book, it should be less than 2 lines.
>>>
>> I have numerous books on logic. It is more than two lines in the few I
>> looked at before answering.
>> -- 
>> The world will little note, nor long remember what we say here
>> Abraham Lincoln at Gettysburg
> 
> Yes, I found several plausible ones on internet.
> But, which one can be used to define irrational numbers better?
> 

Hold on a minute!  It is not the job of logic to define irrational 
numbers!  There is a famous book by Landau that does that.  (And many an 
algebra text...)

-- 
The world will little note, nor long remember what we say here
Abraham Lincoln at Gettysburg

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#36665

wij <wyniijj@gmail.com> writes:

> On Monday, 19 July 2021 at 23:06:18 UTC+8, Ben Bacarisse wrote:
>> wij <wyn...@gmail.com> writes: 
>> 
>> > Definition of equal sign '=' (mathematics ): 
>> > 1. Basic definition: a=b iff a and b are exchangeable 
>> > 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable 
>> > 
>> > Any suggestion improving the definition of '=' in mathematical 
>> > expression?
>> It's a big topic, but any book on formal logic will explain the 
>> definition used in first order theories. You'll also find an 
>> explanation of bound and free variables which is key to correctly 
>> defining substitution in formulas. 
>> 
> I quickly scanned all possible books related to your "formal logic",...none.
> The '=' is basically used unexplained.

I suggest a slower scan.

-- 
Ben.

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#36679

On Tuesday, 20 July 2021 at 04:45:51 UTC+8, Ben Bacarisse wrote:
> wij <wyn...@gmail.com> writes: 
> 
> > On Monday, 19 July 2021 at 23:06:18 UTC+8, Ben Bacarisse wrote: 
> >> wij <wyn...@gmail.com> writes: 
> >> 
> >> > Definition of equal sign '=' (mathematics ): 
> >> > 1. Basic definition: a=b iff a and b are exchangeable 
> >> > 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable 
> >> > 
> >> > Any suggestion improving the definition of '=' in mathematical 
> >> > expression? 
> >> It's a big topic, but any book on formal logic will explain the 
> >> definition used in first order theories. You'll also find an 
> >> explanation of bound and free variables which is key to correctly 
> >> defining substitution in formulas. 
> >>
> > I quickly scanned all possible books related to your "formal logic",...none. 
> > The '=' is basically used unexplained.
> I suggest a slower scan. 
> 
> -- 
> Ben.

Show what's in your book, it should be less than 2 lines. (prove your saying)

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#36690

wij <wyniijj@gmail.com> writes:

> On Tuesday, 20 July 2021 at 04:45:51 UTC+8, Ben Bacarisse wrote:
>> wij <wyn...@gmail.com> writes: 
>> 
>> > On Monday, 19 July 2021 at 23:06:18 UTC+8, Ben Bacarisse wrote: 
>> >> wij <wyn...@gmail.com> writes: 
>> >> 
>> >> > Definition of equal sign '=' (mathematics ): 
>> >> > 1. Basic definition: a=b iff a and b are exchangeable 
>> >> > 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable 
>> >> > 
>> >> > Any suggestion improving the definition of '=' in mathematical 
>> >> > expression? 
>> >> It's a big topic, but any book on formal logic will explain the 
>> >> definition used in first order theories. You'll also find an 
>> >> explanation of bound and free variables which is key to correctly 
>> >> defining substitution in formulas. 
>> >>
>> > I quickly scanned all possible books related to your "formal logic",...none. 
>> > The '=' is basically used unexplained.
>> I suggest a slower scan. 
>
> Show what's in your book, it should be less than 2 lines. (prove your
> saying)

Why?

-- 
Ben.

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#36854

On Monday, July 19, 2021 at 1:24:49 PM UTC-4, wij wrote:
> On Monday, 19 July 2021 at 23:06:18 UTC+8, Ben Bacarisse wrote: 
> > wij <wyn...@gmail.com> writes: 
> > 
> > > Definition of equal sign '=' (mathematics ): 
> > > 1. Basic definition: a=b iff a and b are exchangeable 
> > > 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable 
> > > 
> > > Any suggestion improving the definition of '=' in mathematical 
> > > expression? 
> > It's a big topic, but any book on formal logic will explain the 
> > definition used in first order theories. You'll also find an 
> > explanation of bound and free variables which is key to correctly 
> > defining substitution in formulas. 
> > 
> > -- 
> > Ben.
> I quickly scanned all possible books related to your "formal logic",...none. 
> The '=' is basically used unexplained. 
> 
> Usage example: (A little different than usual meaning. Just for example, I am 
> talking about how to define '='(equal) in average meaning commonly used) 
> 
> Definition: G=x^2 or G=a&b 
> Modulo: a=b (mod n) 
> 
> Similar ones are: 
> x ⊢ y means y is derivable from x. 
> A ↔ B, A ⇔ B means A is true iff B is true 
> 
> ------------------------- 
> To be more specific about the question. 
> If you are to define '=', how would you address it? or others. 
> 
> Example: 
> 1. lim(n->∞) 1/n=0 
> 2. (a+b)^2=a^2+2ab+b^2
They are equal by definition of the syntax of the 2 operands of =.
Each operand produces the same value by their definitions.

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#36863

On Thursday, 22 July 2021 at 23:26:49 UTC+8, Charlie-Boo wrote:
> On Monday, July 19, 2021 at 1:24:49 PM UTC-4, wij wrote: 
> > On Monday, 19 July 2021 at 23:06:18 UTC+8, Ben Bacarisse wrote: 
> > > wij <wyn...@gmail.com> writes: 
> > > 
> > > > Definition of equal sign '=' (mathematics ): 
> > > > 1. Basic definition: a=b iff a and b are exchangeable 
> > > > 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable 
> > > > 
> > > > Any suggestion improving the definition of '=' in mathematical 
> > > > expression? 
> > > It's a big topic, but any book on formal logic will explain the 
> > > definition used in first order theories. You'll also find an 
> > > explanation of bound and free variables which is key to correctly 
> > > defining substitution in formulas. 
> > > 
> > > -- 
> > > Ben. 
> > I quickly scanned all possible books related to your "formal logic",...none. 
> > The '=' is basically used unexplained. 
> > 
> > Usage example: (A little different than usual meaning. Just for example, I am 
> > talking about how to define '='(equal) in average meaning commonly used) 
> > 
> > Definition: G=x^2 or G=a&b 
> > Modulo: a=b (mod n) 
> > 
> > Similar ones are: 
> > x ⊢ y means y is derivable from x. 
> > A ↔ B, A ⇔ B means A is true iff B is true 
> > 
> > ------------------------- 
> > To be more specific about the question. 
> > If you are to define '=', how would you address it? or others. 
> > 
> > Example: 
> > 1. lim(n->∞) 1/n=0 
> > 2. (a+b)^2=a^2+2ab+b^2
> They are equal by definition of the syntax of the 2 operands of =. 
> Each operand produces the same value by their definitions.

I am considering using the term "substitute". Thus,

x=y iff the occurrence of x can be substituted by y, vice versa.

The problem of  "Each operand produces the same value by their definitions."
is that it contains "value" which is not easy to verify (by program), or maybe redundant.

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#36852

On Monday, July 19, 2021 at 11:06:18 AM UTC-4, Ben Bacarisse wrote:
> wij <wyn...@gmail.com> writes: 
> 
> > Definition of equal sign '=' (mathematics ): 
> > 1. Basic definition: a=b iff a and b are exchangeable 
> > 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable 
> > 
> > Any suggestion improving the definition of '=' in mathematical 
> > expression?
> It's a big topic, but any book on formal logic will explain the 
> definition used in first order theories. You'll also find an 
> explanation of bound and free variables which is key to correctly 
> defining substitution in formulas. 
> 
> -- 
> Ben.
When is it not a license to substitute?  When is a license to substitute not equality?
They and you miss the point.

C-B
A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street. ~ David Hilbert
The definition of genius is taking the complex and making it simple. ~ Albert Einstein

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#36856

On Monday, July 19, 2021 at 11:06:18 AM UTC-4, Ben Bacarisse wrote:
> wij <wyn...@gmail.com> writes: 
> 
> > Definition of equal sign '=' (mathematics ): 
> > 1. Basic definition: a=b iff a and b are exchangeable 
> > 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable 
> > 
> > Any suggestion improving the definition of '=' in mathematical 
> > expression?
> It's a big topic, but any book on formal logic will explain the 
> definition used in first order theories. You'll also find an 
> explanation of bound and free variables which is key to correctly 
> defining substitution in formulas. 
> 
> -- 
> Ben.

When is x=y not the same as x can be substituted for y?

“In questions of science the authority of a thousand is not worth the humble reasoning of a single individual.”
~ Galileo Galilei

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#36858

On Thursday, July 22, 2021 at 11:51:09 AM UTC-4, Charlie-Boo wrote:
> On Monday, July 19, 2021 at 11:06:18 AM UTC-4, Ben Bacarisse wrote:
> > wij <wyn...@gmail.com> writes: 
> > 
> > > Definition of equal sign '=' (mathematics ): 
> > > 1. Basic definition: a=b iff a and b are exchangeable 
> > > 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable 
> > > 
> > > Any suggestion improving the definition of '=' in mathematical 
> > > expression? 
> > It's a big topic, but any book on formal logic will explain the 
> > definition used in first order theories. You'll also find an 
> > explanation of bound and free variables which is key to correctly 
> > defining substitution in formulas. 
> > 
> > -- 
> > Ben.
> When is x=y not the same as x can be substituted for y? 
> 
> “In questions of science the authority of a thousand is not worth the humble reasoning of a single individual.” 
> ~ Galileo Galilei

make that: x can be substituted for y and vice-versa.

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#36653

wij wrote:
> Definition of equal sign '=' (mathematics ):
> 1. Basic definition: a=b iff a and b are exchangeable
> 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable
> 
> Any suggestion improving the definition of '=' in mathematical expression?
> (The axiom of extension in set theory is not appropriate, because it is set only)
> 

If everything is a set then "set only" is not a limitation!

One definition of identity is that two things are identical if 
everything true of one is true of the the and vice versa.

Substitution can be a problem.  Consider -

(i) The morning star is identical to the evening star. (True, for both 
are the plant Venus.)
(ii) Fred believes that the morning star is Venus.

 From those, if exchangeability holds, one could conclude Fred believes 
that the evening star is Venus.  But that may not be so.

-- 
The world will little note, nor long remember what we say here
Abraham Lincoln at Gettysburg

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#36677

On Tuesday, 20 July 2021 at 01:28:34 UTC+8, Peter wrote:
> wij wrote: 
> > Definition of equal sign '=' (mathematics ): 
> > 1. Basic definition: a=b iff a and b are exchangeable 
> > 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable 
> > 
> > Any suggestion improving the definition of '=' in mathematical expression? 
> > (The axiom of extension in set theory is not appropriate, because it is set only) 
> >
> If everything is a set then "set only" is not a limitation! 
 
Illustrate the premises "everything is a set", in particular numbers,
if it is not empty words.

> One definition of identity is that two things are identical if 
> everything true of one is true of the the and vice versa. 
> 
> Substitution can be a problem. Consider - 
> 
> (i) The morning star is identical to the evening star. (True, for both 
> are the plant Venus.) 
> (ii) Fred believes that the morning star is Venus. 
> 
> From those, if exchangeability holds, one could conclude Fred believes 
> that the evening star is Venus. But that may not be so. 
> 
> -- 
> The world will little note, nor long remember what we say here 
> Abraham Lincoln at Gettysburg

I do not see '=' and its explanation.

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#36855

On Monday, July 19, 2021 at 9:27:17 PM UTC-4, wij wrote:
> On Tuesday, 20 July 2021 at 01:28:34 UTC+8, Peter wrote: 
> > wij wrote: 
> > > Definition of equal sign '=' (mathematics ): 
> > > 1. Basic definition: a=b iff a and b are exchangeable 
> > > 2. For any? equation X=Y, the occurrence of a,b in X,Y are exchangeable 
> > > 
> > > Any suggestion improving the definition of '=' in mathematical expression? 
> > > (The axiom of extension in set theory is not appropriate, because it is set only) 
> > > 
> > If everything is a set then "set only" is not a limitation!
> Illustrate the premises "everything is a set", in particular numbers, 
> if it is not empty words.
> > One definition of identity is that two things are identical if 
> > everything true of one is true of the the and vice versa. 
> > 
> > Substitution can be a problem. Consider - 
> > 
> > (i) The morning star is identical to the evening star. (True, for both 
> > are the plant Venus.) 
> > (ii) Fred believes that the morning star is Venus. 
> > 
> > From those, if exchangeability holds, one could conclude Fred believes 
> > that the evening star is Venus. But that may not be so. 
> > 
> > -- 
> > The world will little note, nor long remember what we say here 
> > Abraham Lincoln at Gettysburg
> I do not see '=' and its explanation.
"everything is a set" is not to be taken literally - it is just an idiomatic expression.
You are not a set.  You are a person.

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