What are the base assumptions we make in mathematics?
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In any proof, we establish (or attempt to establish) a fact based on previously established facts. Each of these previously established facts in turn must have been established using a set of then pre-established facts, and so on and so forth until either:
- An infinite sequence of regressive proofs is reached, which is unlikely, as that would require an infinite body of mathematics just to have reached the point we are at now, considering every fact now by definition above would be built upon this infinite stack of facts, or...
- We reach our base facts which are assumed or used as defining math itself, upon the foundation of which all other facts are proven
So what are these base facts underlying mathematics? Are they defined somewhere?
logic soft-question axioms
edited Nov 20 at 15:26
Mauro ALLEGRANZA
63.4k448110
asked Nov 20 at 15:23
TheEnvironmentalist
18210
add a comment |
up vote
0
down vote
favorite
In any proof, we establish (or attempt to establish) a fact based on previously established facts. Each of these previously established facts in turn must have been established using a set of then pre-established facts, and so on and so forth until either:
- An infinite sequence of regressive proofs is reached, which is unlikely, as that would require an infinite body of mathematics just to have reached the point we are at now, considering every fact now by definition above would be built upon this infinite stack of facts, or...
- We reach our base facts which are assumed or used as defining math itself, upon the foundation of which all other facts are proven
So what are these base facts underlying mathematics? Are they defined somewhere?
logic soft-question axioms
edited Nov 20 at 15:26
Mauro ALLEGRANZA
63.4k448110
asked Nov 20 at 15:23
TheEnvironmentalist
18210
What you want to look into is the concept of "axioms".
– Eff
Nov 20 at 15:25
NO "infinite sequence of regressive proofs is reached".
– Mauro ALLEGRANZA
Nov 20 at 15:25
"We reach our base facts which are assumed" : Axioms.
– Mauro ALLEGRANZA
Nov 20 at 15:26
1
In addition to the axioms of set theory, we usually assume some basic logic, which is essential to allow us to write proofs in the first place. But the key thing is "axioms," and most of the ones we assume are discussed in set theory. I've found Halmos's book "Naive Set Theory" to be a fine place to start learning about these most basic axioms.
– John Hughes
Nov 20 at 15:40
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In any proof, we establish (or attempt to establish) a fact based on previously established facts. Each of these previously established facts in turn must have been established using a set of then pre-established facts, and so on and so forth until either:
- An infinite sequence of regressive proofs is reached, which is unlikely, as that would require an infinite body of mathematics just to have reached the point we are at now, considering every fact now by definition above would be built upon this infinite stack of facts, or...
- We reach our base facts which are assumed or used as defining math itself, upon the foundation of which all other facts are proven
So what are these base facts underlying mathematics? Are they defined somewhere?
logic soft-question axioms
edited Nov 20 at 15:26
Mauro ALLEGRANZA
63.4k448110
asked Nov 20 at 15:23
TheEnvironmentalist
18210
In any proof, we establish (or attempt to establish) a fact based on previously established facts. Each of these previously established facts in turn must have been established using a set of then pre-established facts, and so on and so forth until either:
- An infinite sequence of regressive proofs is reached, which is unlikely, as that would require an infinite body of mathematics just to have reached the point we are at now, considering every fact now by definition above would be built upon this infinite stack of facts, or...
- We reach our base facts which are assumed or used as defining math itself, upon the foundation of which all other facts are proven
So what are these base facts underlying mathematics? Are they defined somewhere?
logic soft-question axioms
logic soft-question axioms
edited Nov 20 at 15:26
Mauro ALLEGRANZA
63.4k448110
asked Nov 20 at 15:23
TheEnvironmentalist
18210
edited Nov 20 at 15:26
Mauro ALLEGRANZA
63.4k448110
asked Nov 20 at 15:23
TheEnvironmentalist
18210
edited Nov 20 at 15:26
Mauro ALLEGRANZA
63.4k448110
edited Nov 20 at 15:26
Mauro ALLEGRANZA
63.4k448110
edited Nov 20 at 15:26
Mauro ALLEGRANZA
63.4k448110
63.4k448110
asked Nov 20 at 15:23
TheEnvironmentalist
18210
asked Nov 20 at 15:23
TheEnvironmentalist
18210
asked Nov 20 at 15:23
TheEnvironmentalist
18210
18210
What you want to look into is the concept of "axioms".
– Eff
Nov 20 at 15:25
NO "infinite sequence of regressive proofs is reached".
– Mauro ALLEGRANZA
Nov 20 at 15:25
"We reach our base facts which are assumed" : Axioms.
– Mauro ALLEGRANZA
Nov 20 at 15:26
1
In addition to the axioms of set theory, we usually assume some basic logic, which is essential to allow us to write proofs in the first place. But the key thing is "axioms," and most of the ones we assume are discussed in set theory. I've found Halmos's book "Naive Set Theory" to be a fine place to start learning about these most basic axioms.
– John Hughes
Nov 20 at 15:40
add a comment |
What you want to look into is the concept of "axioms".
– Eff
Nov 20 at 15:25
NO "infinite sequence of regressive proofs is reached".
– Mauro ALLEGRANZA
Nov 20 at 15:25
"We reach our base facts which are assumed" : Axioms.
– Mauro ALLEGRANZA
Nov 20 at 15:26
1
In addition to the axioms of set theory, we usually assume some basic logic, which is essential to allow us to write proofs in the first place. But the key thing is "axioms," and most of the ones we assume are discussed in set theory. I've found Halmos's book "Naive Set Theory" to be a fine place to start learning about these most basic axioms.
– John Hughes
Nov 20 at 15:40
What you want to look into is the concept of "axioms".
– Eff
Nov 20 at 15:25
What you want to look into is the concept of "axioms".
– Eff
Nov 20 at 15:25
NO "infinite sequence of regressive proofs is reached".
– Mauro ALLEGRANZA
Nov 20 at 15:25
NO "infinite sequence of regressive proofs is reached".
– Mauro ALLEGRANZA
Nov 20 at 15:25
"We reach our base facts which are assumed" : Axioms.
– Mauro ALLEGRANZA
Nov 20 at 15:26
"We reach our base facts which are assumed" : Axioms.
– Mauro ALLEGRANZA
Nov 20 at 15:26
1
1
In addition to the axioms of set theory, we usually assume some basic logic, which is essential to allow us to write proofs in the first place. But the key thing is "axioms," and most of the ones we assume are discussed in set theory. I've found Halmos's book "Naive Set Theory" to be a fine place to start learning about these most basic axioms.
– John Hughes
Nov 20 at 15:40
In addition to the axioms of set theory, we usually assume some basic logic, which is essential to allow us to write proofs in the first place. But the key thing is "axioms," and most of the ones we assume are discussed in set theory. I've found Halmos's book "Naive Set Theory" to be a fine place to start learning about these most basic axioms.
– John Hughes
Nov 20 at 15:40
add a comment |
1 Answer
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Usually, Zermelo–Fraenkel axioms.
answered Nov 20 at 15:34
Federico
2,252510
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
Usually, Zermelo–Fraenkel axioms.
answered Nov 20 at 15:34
Federico
2,252510
add a comment |
up vote
2
down vote
Usually, Zermelo–Fraenkel axioms.
answered Nov 20 at 15:34
Federico
2,252510
add a comment |
up vote
2
down vote
up vote
2
down vote
Usually, Zermelo–Fraenkel axioms.
answered Nov 20 at 15:34
Federico
2,252510
Usually, Zermelo–Fraenkel axioms.
answered Nov 20 at 15:34
Federico
2,252510
answered Nov 20 at 15:34
Federico
2,252510
answered Nov 20 at 15:34
Federico
2,252510
answered Nov 20 at 15:34
Federico
2,252510
2,252510
add a comment |
add a comment |
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What you want to look into is the concept of "axioms".
– Eff
Nov 20 at 15:25
NO "infinite sequence of regressive proofs is reached".
– Mauro ALLEGRANZA
Nov 20 at 15:25
"We reach our base facts which are assumed" : Axioms.
– Mauro ALLEGRANZA
Nov 20 at 15:26
1
In addition to the axioms of set theory, we usually assume some basic logic, which is essential to allow us to write proofs in the first place. But the key thing is "axioms," and most of the ones we assume are discussed in set theory. I've found Halmos's book "Naive Set Theory" to be a fine place to start learning about these most basic axioms.
– John Hughes
Nov 20 at 15:40