The present paper intends to report two results. It is shown that the formula

<math id="M1"><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>∀</mo><mi mathvariant="normal">y</mi><mo>∀</mo><mi mathvariant="normal">z</mi><mo stretchy="false">[</mo><mo>¬</mo><mi>G</mi><mfenced separators="|"><mrow><mi>x</mi><mo>

</mo><mtext> </mtext><mi>y</mi></mrow></mfenced><mo>→</mo><mo>¬</mo><mi>M</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=64961105&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=64961102&type=

45.12733078

3.89466691

prov

ides the logic underlying gauge symmetry

where

<math id="M2"><mi>M</mi></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=64961109&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=64961106&type=

2.53999996

2.28600001

denotes the predicate of being massive. For the logic of spontaneous symmetry breaking

by Higgs mechanism

we have

<math id="M3"><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>∀</mo><mi mathvariant="normal">y</mi><mo>∀</mo><mi mathvariant="normal">z</mi><mo stretchy="false">[</mo><mi>G</mi><mfenced separators="|"><mrow><mi>x</mi><mo>

</mo><mtext> </mtext><mi>y</mi></mrow></mfenced><mo>→</mo><mi>M</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=64961095&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=64961092&type=

41.40200043

3.89466691

. Notice that the above two formulas are not logically equivalent. The results are obtained by integrating four components

namely

gauge symmetry and Higgs mechanism in quantum field theory

and Gödel's incompleteness theorem and Tarski's indefinability theorem in mathematical logic. Gödel numbering

is the key for arithmetic modeling applied in this paper.