Elements of Set Theory: Recursion on Natural Numbers

Consider the situation where I want you to guess the function such that I only give you two information; that is,

<ol> <li>a starting value <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmYz2JsWcL8xxxs8a3K1cFC23D9vExwexc7GWLKdmRm1hN/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmYz2JsWcL8xxxs8a3K1cFC23D9vExwexc7GWLKdmRm1hN/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmYz2JsWcL8xxxs8a3K1cFC23D9vExwexc7GWLKdmRm1hN/image.png 2x" /> <li>a function<img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmUFp1iscZde592oraZmryaprk8tcxHEP1PyZEuuLBDLde/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmUFp1iscZde592oraZmryaprk8tcxHEP1PyZEuuLBDLde/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmUFp1iscZde592oraZmryaprk8tcxHEP1PyZEuuLBDLde/image.png 2x" /> such that <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmdv7tjBz2mXhfKso2B37MjiUxAhQqVYHzU9KFbcHeG7uK/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmdv7tjBz2mXhfKso2B37MjiUxAhQqVYHzU9KFbcHeG7uK/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmdv7tjBz2mXhfKso2B37MjiUxAhQqVYHzU9KFbcHeG7uK/image.png 2x" /> for all <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQMdntpGeTFL9yw5ARaxB1fsUHP3sfQReD4mgg31W8EhV/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQMdntpGeTFL9yw5ARaxB1fsUHP3sfQReD4mgg31W8EhV/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmQMdntpGeTFL9yw5ARaxB1fsUHP3sfQReD4mgg31W8EhV/image.png 2x" /> <p dir="auto">This gives away all the information where you can compute successively,<br /> <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmZBnyitNmSe9bftbLcDVffYUm2a9gWSgRZdHnW2vfzRKX/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmZBnyitNmSe9bftbLcDVffYUm2a9gWSgRZdHnW2vfzRKX/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmZBnyitNmSe9bftbLcDVffYUm2a9gWSgRZdHnW2vfzRKX/image.png 2x" /> <p dir="auto">Now for a harder question: Suppose we are given a set <em>A, an element <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmRVPxxhgniWojj6cTfQYhjnvbzPayVUo5CNUknkqUxZsT/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmRVPxxhgniWojj6cTfQYhjnvbzPayVUo5CNUknkqUxZsT/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmRVPxxhgniWojj6cTfQYhjnvbzPayVUo5CNUknkqUxZsT/image.png 2x" />, and a function <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmUH7ahkboqi8bxxiVU5jtNUVdcEPJNF8fCLJYPoQ6zZFj/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmUH7ahkboqi8bxxiVU5jtNUVdcEPJNF8fCLJYPoQ6zZFj/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmUH7ahkboqi8bxxiVU5jtNUVdcEPJNF8fCLJYPoQ6zZFj/image.png 2x" />, then how can we show that there exists a function <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmfNNMFYvtx4K1EJA3m7LAkm3hdBLY2HF4Lie6yNTA95nh/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmfNNMFYvtx4K1EJA3m7LAkm3hdBLY2HF4Lie6yNTA95nh/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmfNNMFYvtx4K1EJA3m7LAkm3hdBLY2HF4Lie6yNTA95nh/image.png 2x" /> such that <ol> <li><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmYjyq3dtDAfH2rojTEPiWuBp8bmvjm3e1E7QLpbVxt76v/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmYjyq3dtDAfH2rojTEPiWuBp8bmvjm3e1E7QLpbVxt76v/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmYjyq3dtDAfH2rojTEPiWuBp8bmvjm3e1E7QLpbVxt76v/image.png 2x" /> <li><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmNTNSFTSbzmjPgtixc2ZNrfrDyyorXZ9xazBJWkEsHCFN/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmNTNSFTSbzmjPgtixc2ZNrfrDyyorXZ9xazBJWkEsHCFN/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmNTNSFTSbzmjPgtixc2ZNrfrDyyorXZ9xazBJWkEsHCFN/image.png 2x" /> <p dir="auto">We can prove that there exists a set <em>h that is a function meeting the above conditions. <p dir="auto"><code>Recursion Theorem on<img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmWTmrH7nG8sfra668GvvbiNhQTjn8H57DywmZG4sr7RUp/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmWTmrH7nG8sfra668GvvbiNhQTjn8H57DywmZG4sr7RUp/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmWTmrH7nG8sfra668GvvbiNhQTjn8H57DywmZG4sr7RUp/image.png 2x" /> Let <em>A be a set, <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmaCsaBEiTZXELnvJ2eBKEzRcEnPmJjSC8C99HTVn19rik/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmaCsaBEiTZXELnvJ2eBKEzRcEnPmJjSC8C99HTVn19rik/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmaCsaBEiTZXELnvJ2eBKEzRcEnPmJjSC8C99HTVn19rik/image.png 2x" /> and <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmTCVEYLVQJvhkhxmfWkKsukaWBssht7SWWvNkQ2N6GnGs/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmTCVEYLVQJvhkhxmfWkKsukaWBssht7SWWvNkQ2N6GnGs/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmTCVEYLVQJvhkhxmfWkKsukaWBssht7SWWvNkQ2N6GnGs/image.png 2x" />. Then there exists a unique function <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmTZVvosu6CCzfX1uV9u9ACNyqWfndrDpx6Wzja1eZ5fwx/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmTZVvosu6CCzfX1uV9u9ACNyqWfndrDpx6Wzja1eZ5fwx/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmTZVvosu6CCzfX1uV9u9ACNyqWfndrDpx6Wzja1eZ5fwx/image.png 2x" /> such that<br /> <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQBv2dZ4A6TmiDHAveNCsMDKhdKtnqJ3wJqN3yxdnPhoQ/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQBv2dZ4A6TmiDHAveNCsMDKhdKtnqJ3wJqN3yxdnPhoQ/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmQBv2dZ4A6TmiDHAveNCsMDKhdKtnqJ3wJqN3yxdnPhoQ/image.png 2x" /><br /> and for every <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmdPGPb59N4jt2KLv1YRX7GikTR5BUpHDHwEqnKWZXWMTe/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmdPGPb59N4jt2KLv1YRX7GikTR5BUpHDHwEqnKWZXWMTe/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmdPGPb59N4jt2KLv1YRX7GikTR5BUpHDHwEqnKWZXWMTe/image.png 2x" /><br /> <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmRWg6t355LCMQTr94niYVV8GswZ8ngexJZfLwgWVRf2VV/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmRWg6t355LCMQTr94niYVV8GswZ8ngexJZfLwgWVRf2VV/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmRWg6t355LCMQTr94niYVV8GswZ8ngexJZfLwgWVRf2VV/image.png 2x" /> <hr /> <p dir="auto"><code>Proof <p dir="auto">First, we will let <em>h be the union of many approximating functions. For the purpose of this proof, call a function <em>v acceptable iff <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmTUQnd1aMXJgemvevPHPY6hSQustzwontTa8N5AdKb2xZ/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmTUQnd1aMXJgemvevPHPY6hSQustzwontTa8N5AdKb2xZ/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmTUQnd1aMXJgemvevPHPY6hSQustzwontTa8N5AdKb2xZ/image.png 2x" />, and the following conditions hold: <p dir="auto">(i) If <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmT42jZwtinyTdc6DEoznKuHKsLVofLq9172mSunsViv6G/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmT42jZwtinyTdc6DEoznKuHKsLVofLq9172mSunsViv6G/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmT42jZwtinyTdc6DEoznKuHKsLVofLq9172mSunsViv6G/image.png 2x" /> <p dir="auto">(ii) If <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmZQMFNGSUsoEanGymK8ii158N5yJhcPFt2hwKBHCwM3PM/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmZQMFNGSUsoEanGymK8ii158N5yJhcPFt2hwKBHCwM3PM/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmZQMFNGSUsoEanGymK8ii158N5yJhcPFt2hwKBHCwM3PM/image.png 2x" /> then also <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmcrYbkG6uL91wH2WN6DRX4V1saDDcok5hz9HWd4gcZTPy/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmcrYbkG6uL91wH2WN6DRX4V1saDDcok5hz9HWd4gcZTPy/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmcrYbkG6uL91wH2WN6DRX4V1saDDcok5hz9HWd4gcZTPy/image.png 2x" /> <p dir="auto">Let <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQ2YnTjNS1gnE3EY7Y43K34Vnw8qVwvZBZBT3jvfaBJFF/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQ2YnTjNS1gnE3EY7Y43K34Vnw8qVwvZBZBT3jvfaBJFF/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmQ2YnTjNS1gnE3EY7Y43K34Vnw8qVwvZBZBT3jvfaBJFF/image.png 2x" /> be the set of all acceptable functions and the union of this set is the <em>h function: <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmeUwRf1LcyhNfn1CRzDWz9EKknB3MjvKkLkQBg7ty6ywt/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmeUwRf1LcyhNfn1CRzDWz9EKknB3MjvKkLkQBg7ty6ywt/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmeUwRf1LcyhNfn1CRzDWz9EKknB3MjvKkLkQBg7ty6ywt/image.png 2x" /> <p dir="auto">We claim that this <em>h meets the demands of the theorem, breaking it down into four parts. <ol> <li><em>h is a function <li><em>h is acceptable <li>dom <em>h is all of <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmWHW3e2WzHtoY5pWsCfSP5oEkgaw7JyttUU1zWautrW51/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmWHW3e2WzHtoY5pWsCfSP5oEkgaw7JyttUU1zWautrW51/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmWHW3e2WzHtoY5pWsCfSP5oEkgaw7JyttUU1zWautrW51/image.png 2x" /> <li><em>h is unique <p dir="auto"><code>Example Let <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmNbrKuZwgGHwEmmTEaGXQsbgnxTwkAXchdP4M9wqxTuoc/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmNbrKuZwgGHwEmmTEaGXQsbgnxTwkAXchdP4M9wqxTuoc/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmNbrKuZwgGHwEmmTEaGXQsbgnxTwkAXchdP4M9wqxTuoc/image.png 2x" /> be the set of all integers, positive, negative, and zero:<br /> <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmSRfTVahZjKnJBpL9Rh5zjKhbbzbpr2GUocqT8Pwie2Qw/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmSRfTVahZjKnJBpL9Rh5zjKhbbzbpr2GUocqT8Pwie2Qw/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmSRfTVahZjKnJBpL9Rh5zjKhbbzbpr2GUocqT8Pwie2Qw/image.png 2x" /> <p dir="auto">There is no function <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmbTJn5FwhWerYg7AjvQits5m81igrQbnxNDMadysHCPYy/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmbTJn5FwhWerYg7AjvQits5m81igrQbnxNDMadysHCPYy/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmbTJn5FwhWerYg7AjvQits5m81igrQbnxNDMadysHCPYy/image.png 2x" /> such that for every <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQSHfwpcaKbVRDUinYKZwMYNM6j8ZWk5Bk5NtPTHWJwLS/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmQSHfwpcaKbVRDUinYKZwMYNM6j8ZWk5Bk5NtPTHWJwLS/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmQSHfwpcaKbVRDUinYKZwMYNM6j8ZWk5Bk5NtPTHWJwLS/image.png 2x" />,<br /> <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmUrKpoZEyFuXgHMA72QGwys5YeQoeWqAYVybPcosB9TE8/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmUrKpoZEyFuXgHMA72QGwys5YeQoeWqAYVybPcosB9TE8/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmUrKpoZEyFuXgHMA72QGwys5YeQoeWqAYVybPcosB9TE8/image.png 2x" /> <p dir="auto">If you notice,<br /> <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmbQUdiPPFMSgjLYFPtr2EuDBmrnuZFwtsgV3dQDvMgd13/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmbQUdiPPFMSgjLYFPtr2EuDBmrnuZFwtsgV3dQDvMgd13/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmbQUdiPPFMSgjLYFPtr2EuDBmrnuZFwtsgV3dQDvMgd13/image.png 2x" /> <p dir="auto"><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmagK5bDYzz8rcy5au9YgANW7jtNhY8FgK81j3kdvcRdCG/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmagK5bDYzz8rcy5au9YgANW7jtNhY8FgK81j3kdvcRdCG/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmagK5bDYzz8rcy5au9YgANW7jtNhY8FgK81j3kdvcRdCG/image.png 2x" /> has no starting point similar to the 0 starting point of the recursion on <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmRKtBoygcSjCpbDPLLioQFyPTWpXy8thhhqGdDhS6vX4k/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmRKtBoygcSjCpbDPLLioQFyPTWpXy8thhhqGdDhS6vX4k/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmRKtBoygcSjCpbDPLLioQFyPTWpXy8thhhqGdDhS6vX4k/image.png 2x" />. <hr /> <p dir="auto">Our first application of the recursion theorem will be to show that any Peano system is "just like" <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmc9Lb7ZQ7uMv8U5YN9pCc9f81wuKb3CpfWbWfk4TnpcUx/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmc9Lb7ZQ7uMv8U5YN9pCc9f81wuKb3CpfWbWfk4TnpcUx/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmc9Lb7ZQ7uMv8U5YN9pCc9f81wuKb3CpfWbWfk4TnpcUx/image.png 2x" />. There are other Peano systems; for example, let N be the set <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmW6wej6EdNuZZmVfFZPDXAZUutbGC5ewXPwoepvDncLcm/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmW6wej6EdNuZZmVfFZPDXAZUutbGC5ewXPwoepvDncLcm/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmW6wej6EdNuZZmVfFZPDXAZUutbGC5ewXPwoepvDncLcm/image.png 2x" /> of powers of 2, let <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmS3csysSP1pptxkCZ9E2dYm9yT3fXP8K6ftZzKD7sppM2/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmS3csysSP1pptxkCZ9E2dYm9yT3fXP8K6ftZzKD7sppM2/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmS3csysSP1pptxkCZ9E2dYm9yT3fXP8K6ftZzKD7sppM2/image.png 2x" />, and let <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmePhF49DvPWpsvr7ZDFdZwb7qAiPVHLnykfrtzDXbkz3B/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmePhF49DvPWpsvr7ZDFdZwb7qAiPVHLnykfrtzDXbkz3B/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmePhF49DvPWpsvr7ZDFdZwb7qAiPVHLnykfrtzDXbkz3B/image.png 2x" />. Then <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmT5SWZ8omr252C4zhDYSztW2JUHwUsL3QgGANftKJf1Ay/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmT5SWZ8omr252C4zhDYSztW2JUHwUsL3QgGANftKJf1Ay/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmT5SWZ8omr252C4zhDYSztW2JUHwUsL3QgGANftKJf1Ay/image.png 2x" /> is a Peano system. <p dir="auto">The following theorem expresses the structural similarity between this Peano system and the <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVdLt8ukBQQpELRPERmuTu5biusxnhXx5EZovMYkVTnfe/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVdLt8ukBQQpELRPERmuTu5biusxnhXx5EZovMYkVTnfe/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmVdLt8ukBQQpELRPERmuTu5biusxnhXx5EZovMYkVTnfe/image.png 2x" />. <p dir="auto"><code>Theorem 4H Let <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVwvQF2AsZPqUR8DQCnMPZy1hbRUfAjBLZRq3Qwi9GYDr/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVwvQF2AsZPqUR8DQCnMPZy1hbRUfAjBLZRq3Qwi9GYDr/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmVwvQF2AsZPqUR8DQCnMPZy1hbRUfAjBLZRq3Qwi9GYDr/image.png 2x" /> be a Peano system. Then <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmddQvHtrdTzNiwi6bbTtiMNXbCMefHwczRtWvoDjKSd9D/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmddQvHtrdTzNiwi6bbTtiMNXbCMefHwczRtWvoDjKSd9D/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmddQvHtrdTzNiwi6bbTtiMNXbCMefHwczRtWvoDjKSd9D/image.png 2x" /> is isomorphic to <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmXqMTFd81xdwPVm1gNqyVn5JJgTKbeBNPNHtRgKNY2sPc/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmXqMTFd81xdwPVm1gNqyVn5JJgTKbeBNPNHtRgKNY2sPc/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmXqMTFd81xdwPVm1gNqyVn5JJgTKbeBNPNHtRgKNY2sPc/image.png 2x" />, i.e., there is a function <em>h mapping <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmZMihmFFmTkCbMJGSgo4s7MffdNCorvV2rxEhK6vKtzPQ/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmZMihmFFmTkCbMJGSgo4s7MffdNCorvV2rxEhK6vKtzPQ/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmZMihmFFmTkCbMJGSgo4s7MffdNCorvV2rxEhK6vKtzPQ/image.png 2x" /> one-to-one onto <em>N in a way that preserves the successor operation.<br /> <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmcELC8Upe6ZXkvC8wiEofCwn7nYQhdr2KBW2yKDhRLGJp/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmcELC8Upe6ZXkvC8wiEofCwn7nYQhdr2KBW2yKDhRLGJp/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmcELC8Upe6ZXkvC8wiEofCwn7nYQhdr2KBW2yKDhRLGJp/image.png 2x" /> <p dir="auto">and the zero element<br /> <center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVTZZNHdT74GrahWQyyzNkkKVwpZX6n7WwR5qUe4DQ1BA/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmVTZZNHdT74GrahWQyyzNkkKVwpZX6n7WwR5qUe4DQ1BA/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmVTZZNHdT74GrahWQyyzNkkKVwpZX6n7WwR5qUe4DQ1BA/image.png 2x" /> <p dir="auto"><strong>Remark: The equation <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmNPxvms9jAdMbYZyEbot6X2g2LqdzovY5kQjZbdcQdEyC/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmNPxvms9jAdMbYZyEbot6X2g2LqdzovY5kQjZbdcQdEyC/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmNPxvms9jAdMbYZyEbot6X2g2LqdzovY5kQjZbdcQdEyC/image.png 2x" /> together with <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmbzCqbbUrov7w2V64NAc4we4Gw9S8htXx9ve6WmDEQQ9K/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmbzCqbbUrov7w2V64NAc4we4Gw9S8htXx9ve6WmDEQQ9K/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmbzCqbbUrov7w2V64NAc4we4Gw9S8htXx9ve6WmDEQQ9K/image.png 2x" /> implies that <img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmbQKSXv3HCrf99f4NDtmpx1jNaptQPzM5iLNfQxeyymuK/image.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmbQKSXv3HCrf99f4NDtmpx1jNaptQPzM5iLNfQxeyymuK/image.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmbQKSXv3HCrf99f4NDtmpx1jNaptQPzM5iLNfQxeyymuK/image.png 2x" />. This can be shown as follows, <p dir="auto"><center><img src="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmXR5JFM67WRpsTn3VyiHiBWNiofVV4ghZd3r4391Hcgdx/1529365509307.png" alt="1529365509307.png" srcset="https://images.hive.blog/768x0/https://cdn.steemitimages.com/DQmXR5JFM67WRpsTn3VyiHiBWNiofVV4ghZd3r4391Hcgdx/1529365509307.png 1x, https://images.hive.blog/1536x0/https://cdn.steemitimages.com/DQmXR5JFM67WRpsTn3VyiHiBWNiofVV4ghZd3r4391Hcgdx/1529365509307.png 2x" /> <p dir="auto">Theorems 4D and 4H relate the constructive approach to the natural numbers and the axiomatic approach. Theorem 4D shows that Peano's postulates are true of the number system we have constructed. And theorem 4H shows that the number system we have constructed is, "to within isomorphism," the only system satisfying Peano's postulates. <p dir="auto"><br /> <hr /> <sup>Disclaimer: this is a summary of section 4.3 from the book "Elements of Set Theory" by Herbert B. Enderton, the content apart from rephrasing is identical, most of the equations are from the book and the same examples are treated. All of the equation images were screenshots from generated latex form using typora <ol> <li><a href="https://github.com/valjen/book_collection" target="_blank" rel="nofollow noreferrer noopener" title="This link will take you away from hive.blog" class="external_link">Elements of Set Theory by Herbert B. Enderton <p dir="auto"><center><br /> Thank you for reading ...<br /><br /><br /> <span> <img src="https://images.hive.blog/0x0/https://steemitimages.com/0x0/https://media.giphy.com/media/ohdY5OaQmUmVW/giphy.gif" /><span> <img src="https://images.hive.blog/768x0/https://steemitimages.com/DQmNXXLQdbXpB1WuRn2YErhQLznZVCdUW3kc7sutPWxA8vv/image.png" srcset="https://images.hive.blog/768x0/https://steemitimages.com/DQmNXXLQdbXpB1WuRn2YErhQLznZVCdUW3kc7sutPWxA8vv/image.png 1x, https://images.hive.blog/1536x0/https://steemitimages.com/DQmNXXLQdbXpB1WuRn2YErhQLznZVCdUW3kc7sutPWxA8vv/image.png 2x" />