Why $x=x$? Why $x=y$ and $y=z$ imply $x=z$? (Assume, $x$, $y$ and $z$ are reals.)What does it mean to axiomatize a logic?Peano axioms: 3 or 5 axioms?if arithmetic is not axiomatizable, why are the Peano Axioms called so?Do Hilbert axioms for Euclidean geometry uniquely characterize the model?Two types of axiomatic theories in mathematicsCan Peano axioms be used to construct a model of Natural numbers?What is the purpose of axiomatic systems?The axiomatic method to real number system VS the constructive method(genetic method)Do Tarski's Axioms prove all of Euclid's Elements?Why and how is logic related to set theory?
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Why $x=x$? Why $x=y$ and $y=z$ imply $x=z$? (Assume, $x$, $y$ and $z$ are reals.)
What does it mean to axiomatize a logic?Peano axioms: 3 or 5 axioms?if arithmetic is not axiomatizable, why are the Peano Axioms called so?Do Hilbert axioms for Euclidean geometry uniquely characterize the model?Two types of axiomatic theories in mathematicsCan Peano axioms be used to construct a model of Natural numbers?What is the purpose of axiomatic systems?The axiomatic method to real number system VS the constructive method(genetic method)Do Tarski's Axioms prove all of Euclid's Elements?Why and how is logic related to set theory?
$begingroup$
I recently have been studying the axiomatic construction of the set of real numbers through the Peano axioms for the natural numbers. It seems to me, the only things needed to proceed with this body of knowledge, is some basic set theory (ZFC axioms) plus the rules of mathematical logic. However, I noticed that from time to time one needs the facts mentioned in the title. Now, these are self-explanatory and self-evident of course but since I am so deep in the roots of modern mathematics, I thought I should inquire about those as well. After all, Euclid states a similar axiom in his Elements: «if each one of two line segments is equal to a third one, then they are equal.»
So, regarding real numbers, where did these rules come from?
- Are they axioms of some sort and if yes from which theory?
- Are they simply some statement formulas of boolean logic? If so, how
is one to be convinced of their validity?
Thanks so much.
calculus logic real-numbers axioms
edited Mar 17 at 1:00
Jack
27.6k1782203
asked Mar 16 at 23:35
Efthymios TsakalerisEfthymios Tsakaleris
111
$endgroup$
|
show 1 more comment
$begingroup$
I recently have been studying the axiomatic construction of the set of real numbers through the Peano axioms for the natural numbers. It seems to me, the only things needed to proceed with this body of knowledge, is some basic set theory (ZFC axioms) plus the rules of mathematical logic. However, I noticed that from time to time one needs the facts mentioned in the title. Now, these are self-explanatory and self-evident of course but since I am so deep in the roots of modern mathematics, I thought I should inquire about those as well. After all, Euclid states a similar axiom in his Elements: «if each one of two line segments is equal to a third one, then they are equal.»
So, regarding real numbers, where did these rules come from?
- Are they axioms of some sort and if yes from which theory?
- Are they simply some statement formulas of boolean logic? If so, how
is one to be convinced of their validity?
Thanks so much.
calculus logic real-numbers axioms
edited Mar 17 at 1:00
Jack
27.6k1782203
asked Mar 16 at 23:35
Efthymios TsakalerisEfthymios Tsakaleris
111
$endgroup$
$begingroup$
Axioms 2-4 here explicitly assume this.
$endgroup$
– J.G.
Mar 16 at 23:40
$begingroup$
The basic properties of $=$ (and other basic facts) are assumed at the level of first-order logic itself. We can of course consider alternate logical systems, but that's where those basic facts are "built in."
$endgroup$
– Noah Schweber
Mar 16 at 23:42
3
$begingroup$
Since you're building up the real number system from the natural numbers, you must be using some set theory to provide the needed tools for that construction (Dedekind cuts, equivalence classes of Cauchy sequences, etc.). In most modern set theories, equality and its basic properties are taken as given in the underlying logic. In some older set theories, equality was defined: for example, some authors defined $x=y$ to mean $forall z,(zin xiff zin y)$. Then the definition makes it easy to prove basic properties of equality like those you asked about.
$endgroup$
– Andreas Blass
Mar 16 at 23:48
$begingroup$
Just to clarify, is your second question how we can rigorously see that these are theorems, or how one justifies them informally?
$endgroup$
– Malice Vidrine
Mar 17 at 3:35
$begingroup$
@Malice Vidrine rigorously, thanks
$endgroup$
– Efthymios Tsakaleris
Mar 17 at 15:05
|
show 1 more comment
$begingroup$
I recently have been studying the axiomatic construction of the set of real numbers through the Peano axioms for the natural numbers. It seems to me, the only things needed to proceed with this body of knowledge, is some basic set theory (ZFC axioms) plus the rules of mathematical logic. However, I noticed that from time to time one needs the facts mentioned in the title. Now, these are self-explanatory and self-evident of course but since I am so deep in the roots of modern mathematics, I thought I should inquire about those as well. After all, Euclid states a similar axiom in his Elements: «if each one of two line segments is equal to a third one, then they are equal.»
So, regarding real numbers, where did these rules come from?
- Are they axioms of some sort and if yes from which theory?
- Are they simply some statement formulas of boolean logic? If so, how
is one to be convinced of their validity?
Thanks so much.
calculus logic real-numbers axioms
edited Mar 17 at 1:00
Jack
27.6k1782203
asked Mar 16 at 23:35
Efthymios TsakalerisEfthymios Tsakaleris
111
$endgroup$
I recently have been studying the axiomatic construction of the set of real numbers through the Peano axioms for the natural numbers. It seems to me, the only things needed to proceed with this body of knowledge, is some basic set theory (ZFC axioms) plus the rules of mathematical logic. However, I noticed that from time to time one needs the facts mentioned in the title. Now, these are self-explanatory and self-evident of course but since I am so deep in the roots of modern mathematics, I thought I should inquire about those as well. After all, Euclid states a similar axiom in his Elements: «if each one of two line segments is equal to a third one, then they are equal.»
So, regarding real numbers, where did these rules come from?
- Are they axioms of some sort and if yes from which theory?
- Are they simply some statement formulas of boolean logic? If so, how
is one to be convinced of their validity?
Thanks so much.
calculus logic real-numbers axioms
calculus logic real-numbers axioms
edited Mar 17 at 1:00
Jack
27.6k1782203
asked Mar 16 at 23:35
Efthymios TsakalerisEfthymios Tsakaleris
111
edited Mar 17 at 1:00
Jack
27.6k1782203
asked Mar 16 at 23:35
Efthymios TsakalerisEfthymios Tsakaleris
111
edited Mar 17 at 1:00
Jack
27.6k1782203
edited Mar 17 at 1:00
Jack
27.6k1782203
edited Mar 17 at 1:00
Jack
27.6k1782203
27.6k1782203
asked Mar 16 at 23:35
Efthymios TsakalerisEfthymios Tsakaleris
111
asked Mar 16 at 23:35
Efthymios TsakalerisEfthymios Tsakaleris
111
asked Mar 16 at 23:35
Efthymios TsakalerisEfthymios Tsakaleris
111
111
$begingroup$
Axioms 2-4 here explicitly assume this.
$endgroup$
– J.G.
Mar 16 at 23:40
$begingroup$
The basic properties of $=$ (and other basic facts) are assumed at the level of first-order logic itself. We can of course consider alternate logical systems, but that's where those basic facts are "built in."
$endgroup$
– Noah Schweber
Mar 16 at 23:42
3
$begingroup$
Since you're building up the real number system from the natural numbers, you must be using some set theory to provide the needed tools for that construction (Dedekind cuts, equivalence classes of Cauchy sequences, etc.). In most modern set theories, equality and its basic properties are taken as given in the underlying logic. In some older set theories, equality was defined: for example, some authors defined $x=y$ to mean $forall z,(zin xiff zin y)$. Then the definition makes it easy to prove basic properties of equality like those you asked about.
$endgroup$
– Andreas Blass
Mar 16 at 23:48
$begingroup$
Just to clarify, is your second question how we can rigorously see that these are theorems, or how one justifies them informally?
$endgroup$
– Malice Vidrine
Mar 17 at 3:35
$begingroup$
@Malice Vidrine rigorously, thanks
$endgroup$
– Efthymios Tsakaleris
Mar 17 at 15:05
|
show 1 more comment
$begingroup$
Axioms 2-4 here explicitly assume this.
$endgroup$
– J.G.
Mar 16 at 23:40
$begingroup$
The basic properties of $=$ (and other basic facts) are assumed at the level of first-order logic itself. We can of course consider alternate logical systems, but that's where those basic facts are "built in."
$endgroup$
– Noah Schweber
Mar 16 at 23:42
3
$begingroup$
Since you're building up the real number system from the natural numbers, you must be using some set theory to provide the needed tools for that construction (Dedekind cuts, equivalence classes of Cauchy sequences, etc.). In most modern set theories, equality and its basic properties are taken as given in the underlying logic. In some older set theories, equality was defined: for example, some authors defined $x=y$ to mean $forall z,(zin xiff zin y)$. Then the definition makes it easy to prove basic properties of equality like those you asked about.
$endgroup$
– Andreas Blass
Mar 16 at 23:48
$begingroup$
Just to clarify, is your second question how we can rigorously see that these are theorems, or how one justifies them informally?
$endgroup$
– Malice Vidrine
Mar 17 at 3:35
$begingroup$
@Malice Vidrine rigorously, thanks
$endgroup$
– Efthymios Tsakaleris
Mar 17 at 15:05
$begingroup$
Axioms 2-4 here explicitly assume this.
$endgroup$
– J.G.
Mar 16 at 23:40
$begingroup$
Axioms 2-4 here explicitly assume this.
$endgroup$
– J.G.
Mar 16 at 23:40
$begingroup$
The basic properties of $=$ (and other basic facts) are assumed at the level of first-order logic itself. We can of course consider alternate logical systems, but that's where those basic facts are "built in."
$endgroup$
– Noah Schweber
Mar 16 at 23:42
$begingroup$
The basic properties of $=$ (and other basic facts) are assumed at the level of first-order logic itself. We can of course consider alternate logical systems, but that's where those basic facts are "built in."
$endgroup$
– Noah Schweber
Mar 16 at 23:42
3
3
$begingroup$
Since you're building up the real number system from the natural numbers, you must be using some set theory to provide the needed tools for that construction (Dedekind cuts, equivalence classes of Cauchy sequences, etc.). In most modern set theories, equality and its basic properties are taken as given in the underlying logic. In some older set theories, equality was defined: for example, some authors defined $x=y$ to mean $forall z,(zin xiff zin y)$. Then the definition makes it easy to prove basic properties of equality like those you asked about.
$endgroup$
– Andreas Blass
Mar 16 at 23:48
$begingroup$
Since you're building up the real number system from the natural numbers, you must be using some set theory to provide the needed tools for that construction (Dedekind cuts, equivalence classes of Cauchy sequences, etc.). In most modern set theories, equality and its basic properties are taken as given in the underlying logic. In some older set theories, equality was defined: for example, some authors defined $x=y$ to mean $forall z,(zin xiff zin y)$. Then the definition makes it easy to prove basic properties of equality like those you asked about.
$endgroup$
– Andreas Blass
Mar 16 at 23:48
$begingroup$
Just to clarify, is your second question how we can rigorously see that these are theorems, or how one justifies them informally?
$endgroup$
– Malice Vidrine
Mar 17 at 3:35
$begingroup$
Just to clarify, is your second question how we can rigorously see that these are theorems, or how one justifies them informally?
$endgroup$
– Malice Vidrine
Mar 17 at 3:35
$begingroup$
@Malice Vidrine rigorously, thanks
$endgroup$
– Efthymios Tsakaleris
Mar 17 at 15:05
$begingroup$
@Malice Vidrine rigorously, thanks
$endgroup$
– Efthymios Tsakaleris
Mar 17 at 15:05
|
show 1 more comment
1 Answer
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$begingroup$
Are they axioms of some sort and if yes from which theory?
Yhe they are the first-order logic axioms for equlity.
answered Mar 17 at 8:10
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
$endgroup$
add a comment |
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$begingroup$
Are they axioms of some sort and if yes from which theory?
Yhe they are the first-order logic axioms for equlity.
answered Mar 17 at 8:10
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
$endgroup$
add a comment |
$begingroup$
Are they axioms of some sort and if yes from which theory?
Yhe they are the first-order logic axioms for equlity.
answered Mar 17 at 8:10
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
$endgroup$
add a comment |
$begingroup$
Are they axioms of some sort and if yes from which theory?
Yhe they are the first-order logic axioms for equlity.
answered Mar 17 at 8:10
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
$endgroup$
Are they axioms of some sort and if yes from which theory?
Yhe they are the first-order logic axioms for equlity.
answered Mar 17 at 8:10
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
answered Mar 17 at 8:10
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
answered Mar 17 at 8:10
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
answered Mar 17 at 8:10
Mauro ALLEGRANZAMauro ALLEGRANZA
67.4k449116
67.4k449116
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$begingroup$
Axioms 2-4 here explicitly assume this.
$endgroup$
– J.G.
Mar 16 at 23:40
$begingroup$
The basic properties of $=$ (and other basic facts) are assumed at the level of first-order logic itself. We can of course consider alternate logical systems, but that's where those basic facts are "built in."
$endgroup$
– Noah Schweber
Mar 16 at 23:42
3
$begingroup$
Since you're building up the real number system from the natural numbers, you must be using some set theory to provide the needed tools for that construction (Dedekind cuts, equivalence classes of Cauchy sequences, etc.). In most modern set theories, equality and its basic properties are taken as given in the underlying logic. In some older set theories, equality was defined: for example, some authors defined $x=y$ to mean $forall z,(zin xiff zin y)$. Then the definition makes it easy to prove basic properties of equality like those you asked about.
$endgroup$
– Andreas Blass
Mar 16 at 23:48
$begingroup$
Just to clarify, is your second question how we can rigorously see that these are theorems, or how one justifies them informally?
$endgroup$
– Malice Vidrine
Mar 17 at 3:35
$begingroup$
@Malice Vidrine rigorously, thanks
$endgroup$
– Efthymios Tsakaleris
Mar 17 at 15:05