Were there any results that were true in old axioms, but are false in modern axioms, and are not obvious paradoxes? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Results that were widely believed to be false but were later shown to be trueAre there papers or books that explain why Bernhard Riemann believed that his hypothesis is true?Can proof by contradiction 'fail'?What kind of mathematical “discoveries” have enabled mankind to build modern computers?Is it possible to construct a formal system such that all interesting statements from ZFC can be proven within the system?Language of an Axiomatic System in the Incompleteness TheoremAre there any new axioms being developed?Reference request: where is it written explicitly that the paradoxes of the early 20th century were overcame in the current mathematics?On independency of ZFC of statements in math problem solvingHilbert Program: Consistency vs Soundness
Simple Http Server
Asymptotics question
In musical terms, what properties are varied by the human voice to produce different words / syllables?
How does the math work when buying airline miles?
New Order #6: Easter Egg
Central Vacuuming: Is it worth it, and how does it compare to normal vacuuming?
retrieve food groups from food item list
How to force a browser when connecting to a specific domain to be https only using only the client machine?
Found this skink in my tomato plant bucket. Is he trapped? Or could he leave if he wanted?
License to disallow distribution in closed source software, but allow exceptions made by owner?
How many time has Arya actually used Needle?
Positioning dot before text in math mode
Trying to understand entropy as a novice in thermodynamics
One-one communication
Nose gear failure in single prop aircraft: belly landing or nose-gear up landing?
The test team as an enemy of development? And how can this be avoided?
Should a wizard buy fine inks every time he want to copy spells into his spellbook?
Why are vacuum tubes still used in amateur radios?
Why datecode is SO IMPORTANT to chip manufacturers?
Moving a wrapfig vertically to encroach partially on a subsection title
Would color changing eyes affect vision?
Is CEO the "profession" with the most psychopaths?
How to write capital alpha?
Random body shuffle every night—can we still function?
Were there any results that were true in old axioms, but are false in modern axioms, and are not obvious paradoxes?
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Results that were widely believed to be false but were later shown to be trueAre there papers or books that explain why Bernhard Riemann believed that his hypothesis is true?Can proof by contradiction 'fail'?What kind of mathematical “discoveries” have enabled mankind to build modern computers?Is it possible to construct a formal system such that all interesting statements from ZFC can be proven within the system?Language of an Axiomatic System in the Incompleteness TheoremAre there any new axioms being developed?Reference request: where is it written explicitly that the paradoxes of the early 20th century were overcame in the current mathematics?On independency of ZFC of statements in math problem solvingHilbert Program: Consistency vs Soundness
$begingroup$
Recently I learned that it can't be proved that mathematical axioms are consistent. And furthermore, in 1900s math was based on an inconsistent system of axioms.
So, are there any results, that were true in old axioms, were believed to be true (so obvious paradoxes don't count), but are false in modern axioms?
If there are some, then does it mean I can be skeptical about any proved statements, just because we can never be sure that our axioms are consistent?
math-history axioms
asked Apr 2 at 9:07
ArqwerArqwer
61
$endgroup$
add a comment |
$begingroup$
Recently I learned that it can't be proved that mathematical axioms are consistent. And furthermore, in 1900s math was based on an inconsistent system of axioms.
So, are there any results, that were true in old axioms, were believed to be true (so obvious paradoxes don't count), but are false in modern axioms?
If there are some, then does it mean I can be skeptical about any proved statements, just because we can never be sure that our axioms are consistent?
math-history axioms
asked Apr 2 at 9:07
ArqwerArqwer
61
$endgroup$
3
$begingroup$
Where did you learn that?
$endgroup$
– José Carlos Santos
Apr 2 at 9:10
1
$begingroup$
The goal of the modern axiom systems was to get a solid footing under the math that we had already done. Which is to say, the axioms were constructed with the explicit goal of changing as few results as possible. I'm not saying there aren't any, but I think you will have to look hard to find them.
$endgroup$
– Arthur
Apr 2 at 9:10
1
$begingroup$
One particular bit of mathematics, namely set theory, was based on assumptions that we now know to be inconsistent from its beginning in the late 1800s until about the first decade of the 1900s. But that does not infect all of the rest of mathematics, which was at that time done in the same way it had always been, without considering it to be applications of set theory, like some modern descriptions are wont to.
$endgroup$
– Henning Makholm
Apr 2 at 9:13
$begingroup$
@JoséCarlosSantos AFAIK Russel's paradox lead to creation of ZFC, because it has shown inconsistency of axioms that were before ZFC
$endgroup$
– Arqwer
Apr 2 at 9:16
1
$begingroup$
There were no set theory axioms before ZFC.
$endgroup$
– José Carlos Santos
Apr 2 at 9:37
add a comment |
$begingroup$
Recently I learned that it can't be proved that mathematical axioms are consistent. And furthermore, in 1900s math was based on an inconsistent system of axioms.
So, are there any results, that were true in old axioms, were believed to be true (so obvious paradoxes don't count), but are false in modern axioms?
If there are some, then does it mean I can be skeptical about any proved statements, just because we can never be sure that our axioms are consistent?
math-history axioms
asked Apr 2 at 9:07
ArqwerArqwer
61
$endgroup$
Recently I learned that it can't be proved that mathematical axioms are consistent. And furthermore, in 1900s math was based on an inconsistent system of axioms.
So, are there any results, that were true in old axioms, were believed to be true (so obvious paradoxes don't count), but are false in modern axioms?
If there are some, then does it mean I can be skeptical about any proved statements, just because we can never be sure that our axioms are consistent?
math-history axioms
math-history axioms
asked Apr 2 at 9:07
ArqwerArqwer
61
asked Apr 2 at 9:07
ArqwerArqwer
61
asked Apr 2 at 9:07
ArqwerArqwer
61
asked Apr 2 at 9:07
ArqwerArqwer
61
asked Apr 2 at 9:07
ArqwerArqwer
61
61
3
$begingroup$
Where did you learn that?
$endgroup$
– José Carlos Santos
Apr 2 at 9:10
1
$begingroup$
The goal of the modern axiom systems was to get a solid footing under the math that we had already done. Which is to say, the axioms were constructed with the explicit goal of changing as few results as possible. I'm not saying there aren't any, but I think you will have to look hard to find them.
$endgroup$
– Arthur
Apr 2 at 9:10
1
$begingroup$
One particular bit of mathematics, namely set theory, was based on assumptions that we now know to be inconsistent from its beginning in the late 1800s until about the first decade of the 1900s. But that does not infect all of the rest of mathematics, which was at that time done in the same way it had always been, without considering it to be applications of set theory, like some modern descriptions are wont to.
$endgroup$
– Henning Makholm
Apr 2 at 9:13
$begingroup$
@JoséCarlosSantos AFAIK Russel's paradox lead to creation of ZFC, because it has shown inconsistency of axioms that were before ZFC
$endgroup$
– Arqwer
Apr 2 at 9:16
1
$begingroup$
There were no set theory axioms before ZFC.
$endgroup$
– José Carlos Santos
Apr 2 at 9:37
add a comment |
3
$begingroup$
Where did you learn that?
$endgroup$
– José Carlos Santos
Apr 2 at 9:10
1
$begingroup$
The goal of the modern axiom systems was to get a solid footing under the math that we had already done. Which is to say, the axioms were constructed with the explicit goal of changing as few results as possible. I'm not saying there aren't any, but I think you will have to look hard to find them.
$endgroup$
– Arthur
Apr 2 at 9:10
1
$begingroup$
One particular bit of mathematics, namely set theory, was based on assumptions that we now know to be inconsistent from its beginning in the late 1800s until about the first decade of the 1900s. But that does not infect all of the rest of mathematics, which was at that time done in the same way it had always been, without considering it to be applications of set theory, like some modern descriptions are wont to.
$endgroup$
– Henning Makholm
Apr 2 at 9:13
$begingroup$
@JoséCarlosSantos AFAIK Russel's paradox lead to creation of ZFC, because it has shown inconsistency of axioms that were before ZFC
$endgroup$
– Arqwer
Apr 2 at 9:16
1
$begingroup$
There were no set theory axioms before ZFC.
$endgroup$
– José Carlos Santos
Apr 2 at 9:37
3
3
$begingroup$
Where did you learn that?
$endgroup$
– José Carlos Santos
Apr 2 at 9:10
$begingroup$
Where did you learn that?
$endgroup$
– José Carlos Santos
Apr 2 at 9:10
1
1
$begingroup$
The goal of the modern axiom systems was to get a solid footing under the math that we had already done. Which is to say, the axioms were constructed with the explicit goal of changing as few results as possible. I'm not saying there aren't any, but I think you will have to look hard to find them.
$endgroup$
– Arthur
Apr 2 at 9:10
$begingroup$
The goal of the modern axiom systems was to get a solid footing under the math that we had already done. Which is to say, the axioms were constructed with the explicit goal of changing as few results as possible. I'm not saying there aren't any, but I think you will have to look hard to find them.
$endgroup$
– Arthur
Apr 2 at 9:10
1
1
$begingroup$
One particular bit of mathematics, namely set theory, was based on assumptions that we now know to be inconsistent from its beginning in the late 1800s until about the first decade of the 1900s. But that does not infect all of the rest of mathematics, which was at that time done in the same way it had always been, without considering it to be applications of set theory, like some modern descriptions are wont to.
$endgroup$
– Henning Makholm
Apr 2 at 9:13
$begingroup$
One particular bit of mathematics, namely set theory, was based on assumptions that we now know to be inconsistent from its beginning in the late 1800s until about the first decade of the 1900s. But that does not infect all of the rest of mathematics, which was at that time done in the same way it had always been, without considering it to be applications of set theory, like some modern descriptions are wont to.
$endgroup$
– Henning Makholm
Apr 2 at 9:13
$begingroup$
@JoséCarlosSantos AFAIK Russel's paradox lead to creation of ZFC, because it has shown inconsistency of axioms that were before ZFC
$endgroup$
– Arqwer
Apr 2 at 9:16
$begingroup$
@JoséCarlosSantos AFAIK Russel's paradox lead to creation of ZFC, because it has shown inconsistency of axioms that were before ZFC
$endgroup$
– Arqwer
Apr 2 at 9:16
1
1
$begingroup$
There were no set theory axioms before ZFC.
$endgroup$
– José Carlos Santos
Apr 2 at 9:37
$begingroup$
There were no set theory axioms before ZFC.
$endgroup$
– José Carlos Santos
Apr 2 at 9:37
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3171642%2fwere-there-any-results-that-were-true-in-old-axioms-but-are-false-in-modern-axi%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3171642%2fwere-there-any-results-that-were-true-in-old-axioms-but-are-false-in-modern-axi%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
3
$begingroup$
Where did you learn that?
$endgroup$
– José Carlos Santos
Apr 2 at 9:10
1
$begingroup$
The goal of the modern axiom systems was to get a solid footing under the math that we had already done. Which is to say, the axioms were constructed with the explicit goal of changing as few results as possible. I'm not saying there aren't any, but I think you will have to look hard to find them.
$endgroup$
– Arthur
Apr 2 at 9:10
1
$begingroup$
One particular bit of mathematics, namely set theory, was based on assumptions that we now know to be inconsistent from its beginning in the late 1800s until about the first decade of the 1900s. But that does not infect all of the rest of mathematics, which was at that time done in the same way it had always been, without considering it to be applications of set theory, like some modern descriptions are wont to.
$endgroup$
– Henning Makholm
Apr 2 at 9:13
$begingroup$
@JoséCarlosSantos AFAIK Russel's paradox lead to creation of ZFC, because it has shown inconsistency of axioms that were before ZFC
$endgroup$
– Arqwer
Apr 2 at 9:16
1
$begingroup$
There were no set theory axioms before ZFC.
$endgroup$
– José Carlos Santos
Apr 2 at 9:37